MULTIDIMENSIONAL MATRIX INVERSIONS AND MULTIPLE BASIC HYPERGEOMETRIC

SERIES

Dissertation

zur Erlangung des akademischen Grades Dr. rer. nat.

an der Formal- und Naturwissenschaftlichen Fakultät der Universität Wien,

eingereicht von Michael Schlosser Wien, im April 1996

The author was supported by the Austrian Science Foundation FWF, grant P10191-MAT

Author addresses:

Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria.

E-mail address: SCHLOSSE@Pap.Univie.Ac.At

Abstract. We compute the inverse of a specific infinite r-dimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an r-dimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite r-dimensional matrix. As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series. We work in the setting of multiple basic hypergeometric series verywell-poised on the root systems A_{r}, C_{r}, and D_{r}. Our new summation formulas include D_{r} Jackson’s _{8}ϕ_{7} summations, A_{r} and D_{r} quadratic, and D_{r} cubic summations. Further, we derive multivariable generalizations of Bailey’s classical terminating balanced very-well-poised _{10}ϕ_{9} transformation. We obtain C_{r} and D_{r} _{10}ϕ_{9} transformations from an interchange of multisums, combined with A_{r}, C_{r}, and D_{r} extensions of Jackson’s _{8}ϕ_{7} summation. Special cases of our 10ϕ_{9} transformations include multivariable generalizations of Watson’s transformation of an _{8}ϕ_{7} into a multiple of a _{4}ϕ_{3}. We also deduce multidimensional extensions of Sears’ _{4}ϕ_{3} transformation. Furthermore, we derive summation formulas for a different kind of multidimensional basic hypergeometric series associated to root systems of classical type. We proceed by combining the classical one-dimensional summation formulas with certain determinant evaluations. Our theorems include A_{r} extensions of Ramanujan’s bilateral _{1}ψ_{1} sum, C_{r} extensions of Bailey’s very-well-poised _{6}ψ_{6} summation, and a C_{r} extension of Jackson’s very-well-poised _{8}ϕ_{7} summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A_{r}, B_{r}, C_{r}, and D_{r}, respectively, of Chu’s bilateral transformation formula for basic hypergeometric series of Gasper-Karlsson-Minton type. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation and transformation theorems of the classical theory of basic hypergeometric series.

Contents

Chapter I. Introduction 5 Chapter II. An operator method for proving matrix inversions 11 Chapter III. Multidimensional matrix inversions 17 Chapter IV. Multiple basic hypergeometric summation theorems 31 Chapter V. C_{r} and D_{r} very-well-poised _{10}ϕ_{9} transformations 45 Chapter VI. Watson’s and Sears’ transformations 65 Chapter VII. More multidimensional summation theorems 81 Appendix A. Some determinant lemmas 99 Appendix B. Background information 107 Bibliography 111

4 CONTENTS

CHAPTER I Introduction

confugiamus ergo ad series L. Euler

Hypergeometric series are so widespread, that referring to the hypergeometric function of Gauß, W. W. Sawyer [66, p. 63] claimed:

There must be many universities today where 95 percent, if not 100 percent, of the functions studied by physics, engineering, and even mathematics students, are covered by this single symbol F (a, b; c; x).

The analogous theory of q-hypergeometric functions, which arose initially in combinatorics and classical analysis, and contains hypergeometric functions as special cases, interacts similarly with number theory, statistics, physics, and representation theory of quantum Lie algebras.

The purpose of this dissertation is to study multidimensional extensions of some of the fundamental summation and transformation formulas of q-series. While there are several such theories being studied, the one we consider is motivated directly from some work in mathematical physics. Already, in 20 short years, there is evidence to believe that this is a natural generalization of the classical work, with the required depth and rich interaction with other parts of mathematics.

The study of hypergeometric series can be traced back many centuries. Perhaps the most fundamental result in this theory is the famous binomial theorem

(1 + x)^{α} = ∑_{k=0}^{∞} \binom{α}{k}x^{k},

for complex α, which was proved in 1665 by twenty-three year old Newton. The study of basic hypergeometric series (also called q-hypergeometric series or q-series) started essentially in 1748 when Euler [20] considered the infinite product

(q; q)_{∞}^{−1} = ∏_{k=1}^{∞}(1 − q^{k})^{−1}

as a generating function for p(n), the number of partitions of a positive integer n into positive integers.

Euler [21] and Pfaff [62] studied many important properties of the hypergeometric series

2F_{1}[\begin{matrix}a, b \\ c \end{matrix}; z] = ∑_{k=0}^{∞} \frac{(a)_{k}(b)_{k}}{k! (c)_{k}} z^{k},  (I.1)

where the rising factorial (a)_{k} is defined as

(a)_{k} = a(a + 1) ⋯ (a + k − 1).

The hypergeometric series in (I.1) was extensively investigated by Gauß [27] in his doctoral dissertation of 1812, where he proved the following summation formula involving Gamma functions:

2F_{1}[\begin{matrix}a, b \\ c \end{matrix}; 1] = \frac{Γ(c − a − b) Γ(c)}{Γ(c − a) Γ(c − b)},

for integer b ≤ 0 or ℜc > ℜa + ℜb. In 1846 Heine [36] converted the simple observation

lim_{q→1} \frac{1 − q^{a}}{1 − q} = a

into a systematic study of q-hypergeometric series, analogous to the theory of ordinary hypergeometric series. The q-hypergeometric series are usually called basic hypergeometric series, where “basic” refers to the base q. From that time on basic hypergeometric series acquired independent interest, and the pace of progress in the last 150 years has been hectic.

Many important results on hypergeometric and basic hypergeometric series were derived by F. H. Jackson and by W. N. Bailey, and others. The short and brilliant career of Ramanujan, and his beautiful identities, has played its part in attracting many researchers to this area. In more recent years, G. E. Andrews has done much to popularize the area, by explaining the theory and pointing out the applications of q-series. In this dissertation, however, we often refer to Gasper and Rahman’s textbook [25] which has become a standard reference to basic hypergeometric series.

An important feature of hypergeometric and basic hypergeometric series is that a symbolic algebra package can be used to discover as well as check new results. Moreover, algorithms are known (and implemented) for proving identities [76].

However, while we used such packages to discover and check our results, our identities cannot be proved using the known algorithms. This is the case because we are able to find an infinite family of results, only a few special cases of which can be proved using a computer. Our methods here are largely extensions of the classical techniques [25].

The multivariable series we study were introduced by the three mathematical physicists Biedenharn, Holman, and Louck [37] by comparing two ways of computing the matrix elements of multiplicity free Wigner and Racah coefficients for SU(r). Milne [50] found q-analogues of these multiple series identities, and later Gustafson [31] defined multiple basic hypergeometric series associated to other Lie algebras of classical type. These types of multiple series (also called multiple series associated to root systems) are directly related to the corresponding Macdonald identities [51], [31]. Over the last fifteen years the theory of multiple basic hypergeometric series associated to root systems has been extensively developed by Gustafson, Milne, and by their co-workers. Many of the classical (one-dimensional) series identities have been extended to higher dimensions. It is interesting to observe that often many different extensions of the same classical identity exist. Further, it becomes apparent that a lot of facts of the classical theory of basic hypergeometric series are naturally motivated by the multivariable analysis. Thus the study of multiple q-series leads to a better understanding of the classical one-dimensional theory.

Recently, multiple basic hypergeometric series associated to root systems have appeared in combinatorial studies on plane partitions, tableaux, and cylindrical partitions. For example, Krattenthaler [44] has applied such multiple series to compute generating functions for plane partitions and tableaux. Other important work has been done by Gessel and Krattenthaler [28]. This makes it very likely that multiple basic hypergeometric series associated to the root systems will have many more applications in the future.

In this thesis, we derive various new summation and transformation formulas for multiple basic hypergeometric series associated to the root systems A_{r}, C_{r} and D_{r}. One of the main tools that we use for deriving our new summation theorems is matrix inversion. The idea is simple: Suppose, an infinite lower triangular matrix (f_{nk}), and two infinite sequences (a_{k}) and (b_{k}) are given, related by the equation

∑_{0≤k≤n} f_{nk}a_{k} = b_{n}.

If the inverse of (f_{nk}), the infinite lower triangular matrix (g_{kl}), is known, then it is clear that this relation can be inverted, which is

∑_{0≤l≤k} g_{kl}b_{l} = a_{k}.

Matrix inversions are very important in many fields of combinatorics and special functions. When dealing with combinatorial sums, application of matrix inversion may help to simplify problems, or yield new identities. Andrews [4] discovered that the Bailey transform [10], which is a very powerful tool in the theory of (basic) hypergeometric series, corresponds to the inversion of two infinite lower-triangular matrices. Gessel and Stanton [29] used a bibasic extension of that matrix inversion to derive a number of basic hypergeometric summations and transformations, and identities of Rogers-Ramanujan type. Even earlier, Carlitz [17] had found an even more general matrix inversion though without giving any applications.

Gasper and Rahman [24], [63], [26], [25, sec. 3.6] used Gasper’s [24] bibasic matrix inversion together with an indefinite bibasic sum to derive numerous beautiful hypergeometric summation and transformation formulas.

The most general (1-dimensional) matrix inversion, however, which contained all the above inversions, was found by Krattenthaler [45] who applied his inversion to derive a number of hypergeometric summation formulas. The inverse matrices he gave are basically (f_{nk})_{n,k∈ℤ} and (g_{kl})_{k,l∈ℤ} (ℤ denotes the set of integers), where

f_{nk} = \frac{∏_{j=k}^{n−1}(a_{j} − c_{k})}{∏_{j=k+1}^{n}(c_{j} − c_{k})},

and

g_{kl} = \frac{(a_{l} − c_{l})}{(a_{k} − c_{k})} \frac{∏_{j=l+1}^{k}(a_{j} − c_{k})}{∏_{j=l}^{k−1}(c_{j} − c_{k})}.

In fact, the special case a_{j} = aq^{−j}, c_{k} = q^{k} is equivalent to the matrix inversion of Andrews, and the case a_{j} = ap^{−j}, c_{k} = q^{k} is equivalent to Gessel and Stanton’s. Specializing c_{k} = q^{k} we obtain Carlitz’s matrix inversion, and a_{j} = (bp^{−j}/a) + ap^{j}, c_{k} = q^{−k} + bq^{k} yields the inversion of Gasper.

Multidimensional matrix inversions were found by Milne, Lilly and Milne, and by Bhatnagar and Milne. The A_{r} (or equivalently U(r+1)) and C_{r} inversions (corresponding to the root systems A_{r} and C_{r}, respectively) of Milne and Lilly [55, Theorem 3.3], [57], [47], [48], which are higher-dimensional generalizations of Andrews’ Bailey transform matrices, were used to derive A_{r} and C_{r} extensions [55],

[58] of many of the classical hypergeometric summation and transformation formulas. Bhatnagar and Milne [11, Theorem 5.7], [13, Theorem 3.48] were even able to find an A_{r} extension of Gasper’s bibasic hypergeometric matrix inversion. They used a special case of his matrix inversion, an A_{r} extension of Carlitz’s inversion, to derive A_{r} identities of Abel-type. But none of these multidimensional matrix inversions contained Krattenthaler’s inversion as a special case.

One of the main results of this thesis is a multidimensional extension of Krattenthaler’s matrix inverse (see Theorem III.7). This multidimensional matrix inversion unifies all the matrix inversions mentioned so far as it contains them all as special cases. Besides, we present another interesting multidimensional matrix inversion (see Theorem III.29) which is of different type.

In order to prove our matrix inversions in Theorems III.7 and III.29 we utilize Krattenthaler’s operator method [43] which we review in chapter II. We adapt a main theorem of [43] and add an appropriate multidimensional corollary (see Corollary II.18).

The main motivation for finding a multidimensional extension of Krattenthaler’s matrix inverse came from prospective applications to basic hypergeometric series. These applications are the contents of chapter IV. We combine a special case of Theorem III.7 and a C_{r} _{8}ϕ_{7} summation theorem, found independently by Denis and Gustafson [19] and Milne and Lilly [58], to derive a D_{r} _{8}ϕ_{7} summation theorem, which is closely related to a D_{r} summation found by Bhatnagar [12]. We also derive A_{r} and D_{r} extensions of a quadratic hypergeometric summation formula of Gessel and Stanton [29]. Finally, we derive a D_{r} extension of a cubic summation formula of Gasper and Rahman [26]. The above work appears in [67].

Our multidimensional matrix inversions are very useful in the theory of basic hypergeometric series of type A_{r}, C_{r}, and D_{r}, respectively, and lead to the discovery of many more new identities. The D_{r} _{8}ϕ_{7} summation theorems, derived by matrix inversion in chapter IV, lead to new C_{r} and D_{r} extensions of Bailey’s [7] very-well-poised balanced _{10}ϕ_{9} transformation. Previously, Milne and Newcomb [59] have found A_{r} extensions of Bailey’s formula. In chapter V, we use A_{r}, C_{r}, and D_{r} extensions of Jackson’s [40] very-well-poised balanced _{8}ϕ_{7} summation, stated in Appendix B, and an interchange of multisums, to generalize Bailey’s _{10}ϕ_{9} transformation formula to multiple series. As special cases of our _{10}ϕ_{9} transformations, we find several multivariable generalizations of Watson’s [73] transformation of an 8ϕ_{7} into a multiple of a _{4}ϕ_{3}. Further, we deduce multidimensional extensions of Sears’ [69] _{4}ϕ_{3} transformation. Our multiple Watson’s and Sears’ transformations are the contents of chapter VI. Chapter V and chapter VI is joint work with Bhatnagar [14].

Chapter VII can be read independently from the previous ones, but in fact, it is related to our previous work. Here we derive multiple generalizations of many of the classical basic hypergeometric summation formulas. The multiple series appearing in chapter VII were first considered by Gustafson and Krattenthaler [32], [33], who showed how to obtain multivariable summation and transformation formulas from determinant evaluations. The series we consider are different from those appearing elsewhere in this thesis, but they are also associated to root systems of classical type. Our theorems include new A_{r} extensions of Ramanujan’s bilateral _{1}ψ_{1} sum (cf. [35]), C_{r} extensions of Bailey’s very-well-poised _{6}ψ_{6} summation [9], and a C_{r} extension of Jackson’s very-well-poised _{8}ϕ_{7} summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A_{r}, B_{r}, C_{r}, and D_{r}, respectively, of Chu’s [18] bilateral transformation formula for basic hypergeometric series of Gasper-Karlsson-Minton type [23]. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation theorems of the classical theory of basic hypergeometric series. This work will appear in a forthcoming paper [68].

A couple of determinant evaluations are necessary for proving the theorems of this thesis. Two determinant evaluations, which are elegant generalizations of the classical and “symplectic” Vandermonde determinants, turn out to be crucial for our computations in chapter III. Another determinant evaluation is needed in chapter VII. Since these lemmas may have value in a variety of other contexts as well we decided to give them separately in Appendix A.

This thesis has been written under the supervision of C. Krattenthaler. The author feels especially indebted to his supervisor who has patiently provided a lot of help and ideas. Further, it is a pleasure for me to thank G. Bhatnagar for contributing many valuable suggestions, during the period of our joint project on the multiple _{10}ϕ_{9} transformations and their consequences. Finally, I would like to thank my parents and friends for their support of any kind over the last years.

CHAPTER II An operator method for proving matrix inversions

In this chapter we lay the ground work for proving our multidimensional matrix inversions in chapter III. We review Krattenthaler’s operator method [43], adapt a main theorem of [43] and add an appropriate multidimensional corollary (see Corollary II.18).

Let F = (f_{nk})_{n,k∈ℤ^{r}} (as before, ℤ denotes the set of integers) be an infinite lower-triangular r-dimensional matrix; i.e. f_{nk} = 0 unless n ≥ k, by which we mean n_{i} ≥ k_{i} for all i = 1, …, r. The matrix G = (g_{kl})_{k,l∈ℤ} is said to be the inverse matrix of F if and only if

∑_{n≥k≥l} f_{nk}g_{kl} = δ_{nl}

for all n, l ∈ ℤ^{r}, where δ_{nl} is the usual Kronecker delta.

In [43] Krattenthaler gave a method for solving Lagrange inversion problems, which are closely connected with the problem of inverting lower-triangular matrices. We will use his operator method for proving our new theorems in chapter III. By a formal Laurent series we mean a series of the form ∑_{n≥k} a_{n}z^{n}, for some k ∈ ℤ^{r}, where z^{n} = z_{1}^{n_{1}}z_{2}^{n_{2}} ⋯ z_{r}^{n_{r}}. (Note that we actually consider formal Laurent series truncated from below, as in the usual definition the n in the sum runs over all elements of ℤ^{r}.) Given the formal Laurent series a(z) and b(z) we introduce the bilinear form ⟨ , ⟩ by

⟨a(z), b(z)⟩ = ⟨z^{0}⟩(a(z) · b(z)),  (II.1)

where ⟨z^{0}⟩c(z) denotes the coefficient of z^{0} in c(z). Since our Laurent series are truncated from below, (II.1) is well-defined. Given any linear operator L acting on formal Laurent series, L^{∗} denotes the adjoint of L with respect to ⟨ , ⟩; i.e. ⟨La(z), b(z)⟩ = ⟨a(z), L^{∗}b(z)⟩ for all formal Laurent series a(z) and b(z). We need the following special case of [43, Theorem 1].

Lemma II.2 (Krattenthaler). Let F = (f_{nk})_{n,k∈ℤ^{r}} be an infinite lowertriangular r-dimensional matrix with f_{kk} ≠ 0 for all k ∈ ℤ^{r}. For k ∈ ℤ^{r}, define the formal Laurent series f_{k}(z) and g_{k}(z) by f_{k}(z) = ∑_{n≥k} f_{nk}z^{n} and g_{k}(z) = ∑_{l≤k} g_{kl}z^{−l}, where (g_{kl})_{k,l∈ℤ^{r}} is the uniquely determined inverse matrix of F. Suppose that for k ∈ ℤ^{r} a system of equations of the form

U_{j}f_{k}(z) = c_{j}(k)V f_{k}(z),  j = 1, …, r,  (II.3)

holds, where U_{j}, V are linear operators acting on formal Laurent series, V being bijective, and (c_{j}(k))_{k∈ℤ^{r}} are arbitrary sequences of constants. Moreover, we suppose that

for all m, n ∈ ℤ^{r}, m ≠ n, there exists a j with 1 ≤ j ≤ r and c_{j}(m) ≠ c_{j}(n). (II.4)

Then, if h_{k}(z) is a solution of the dual system

U_{j}^{∗}h_{k}(z) = c_{j}(k)V^{∗}h_{k}(z),  j = 1, …, r,  (II.5)

with h_{k}(z) ≢ 0 for all k ∈ ℤ^{r}, the series g_{k}(z) are given by

g_{k}(z) = \frac{1}{⟨f_{k}(z), V^{∗}h_{k}(z)⟩}V^{∗}h_{k}(z).  (II.6)

Proof. Suppose (h_{k}(z))_{k∈ℤ^{r}} satisfies (II.5). By using (II.3) we have

c_{j}(k) ⟨f_{k}(z), V^{∗}h_{n}(z)⟩ =c_{j}(k) ⟨V f_{k}(z), h_{n}(z)⟩ = ⟨U_{j}f_{k}(z), h_{n}(z)⟩
=⟨f_{k}(z), U_{j}^{∗}h_{n}(z)⟩ = c_{j}(n) ⟨f_{k}(z), V^{∗}h_{n}(z)⟩.

From (II.4) this immediately implies ⟨f_{k}(z), V^{∗}h_{n}(z)⟩ = 0 for n ≠ k. For n = k suppose ⟨f_{n}(z), V^{∗}h_{n}(z)⟩ = 0 for some n ∈ ℤ^{r}. Together with the above calculation this means ⟨f_{k}(z), V^{∗}h_{n}(z)⟩ = 0 for all k ∈ ℤ^{r} and we conclude V^{∗}h_{n}(z) ≡ 0. It is easy to show that with our assumptions V^{∗} is bijective, and, therefore, h_{n}(z) ≡ 0 in contradiction to the condition h_{k}(z) ≢ 0 for all k ∈ ℤ^{r}.

Defining g_{k}(z) = ⟨f_{k}(z), V^{∗}h_{k}(z)⟩^{−1}V^{∗}h_{k}(z), we evaluate

⟨f_{k}(z), g_{n}(z)⟩ = \frac{⟨f_{k}(z), V^{∗}h_{n}(z)⟩}{⟨f_{n}(z), V^{∗}h_{n}(z)⟩} = δ_{nk}.

The uniqueness of the sequence (g_{k}(z))_{k∈ℤ^{r}} is obvious.

In our applications we will use a corollary of Lemma II.2 (see Corollary II.18). Let S_{r} be the symmetric group of order r. For possibly noncommuting operators V_{ij} let us define the column determinant by

det^{→}_{1≤i,j≤r}(V_{ij}) := ∑_{σ∈S_{r}} sgn(σ)V_{σ(r),r}V_{σ(r−1),r−1} ⋯ V_{σ(1),1}.  (II.7)

An equivalent, recursive definition is by means of the expansion along the first column,

det^{→}_{1≤i,j≤r}(V_{ij}) = ∑_{k=1}^{r}(−1)^{k+1}V⃗^{(k,1)}V_{k1},  (II.8)

where V⃗^{(i,j)} denotes the column minor with the i-th row and j-th column being omitted.

Proposition II.9. Let (V_{ij})_{i,j=1}^{r} be a matrix of linear operators acting on formal Laurent series. Suppose V_{ij} = C_{ij} + A_{ij} for i, j = 1, …, r, where the operators C_{ij}, A_{ij} obey the following commutation rules

C_{ij}C_{kl} = C_{kl}C_{ij},  i ≠ k; i, j, k, l = 1, …, r,  (II.10)

i.e. C_{ij} and C_{kl} commute when taken from different rows,

C_{ij}A_{kl} = A_{kl}C_{ij},  i ≠ k; i, j, k, l = 1, …, r,  (II.11)

i.e. C_{ij} and A_{kl} commute when taken from different rows, respectively the rules

A_{ij}A_{kl} = A_{kj}A_{il},  i, j, k, l = 1, …, r,  (II.12)

where the column indices j and l keep their order.

Then the column determinant det^{→}_{1≤i,j≤r}(V_{ij}), as defined in (II.7) or (II.8), can be reduced to a polynomial in the A_{ij}’s, i, j = 1, …, r, of degree ≤ 1,

det^{→}_{1≤i,j≤r}(V_{ij}) = det^{→}_{1≤i,j≤r}(C_{ij}) + ∑_{i,j=1}^{r}(−1)^{i+j}C⃗^{(i,j)}A_{ij},  (II.13)

where C⃗^{(i,j)} again denotes the column minor with the i-th row and j-th column being omitted.

Moreover, for any l = 1, …, r the expansion

det^{→}_{1≤i,j≤r}(V_{ij}) = ∑_{k=1}^{r}(−1)^{k+l}V⃗^{(k,l)}V_{kl}  (II.14)

holds, which means that the determinant can be expanded along any arbitrary column.

Proof. Our column determinant can be expanded into

det^{→}_{1≤i,j≤r}(V_{ij}) = _{∑_{σ∈S_{r}} ; q∈{γ,α}^{r}} sgn(σ)X_{σ(r),r}(q_{r}) ⋯ X_{σ(1),1}(q_{1}),  (II.15)

where q = (q_{1}, q_{2}, …, q_{r}) with q_{i} equal to either γ or α for i = 1, …, r, and where X_{ij}(γ) = C_{ij} and X_{ij}(α) = A_{ij}. Due to (II.10) and (II.11), we observe that in every term of (II.15) the X_{ij}(γ) commute with all other variables. X_{ij}(α) and X_{kl}(α) do not commute unless j = l, but due to (II.12) we have X_{ij}(α)X_{kl}(α) = X_{kj}(α)X_{il}(α). This important fact lets all terms in (II.15) cancel where α occurs more than once, since we can pair the terms

sgn(σ) … X_{σ(i),i}(α) … X_{σ(j),j}(α) …

and

sgn(σ · (i, j)) … X_{σ(j),i}(α) … X_{σ(i),j}(α) …,

having chosen the first two occurrences of α, for instance, and where all other factors are unchanged. After cancellation, we are left with terms with at most one occurrence of α and where all factors commute. This implies the first assertion of the proposition.

For proving the second assertion, we define for a given permutation τ ∈ S_{r} a modified column determinant by

det_{1≤i,j≤r}^{→ (τ)}(V_{ij}) = ∑_{σ∈S_{r}} sgn(σ)V_{σ(τ(r)),τ(r)} ⋯ V_{σ(τ(1)),τ(1)},  (II.16)

i.e. the sequence of columns in expanding the determinant is determined by τ.

Expanding det^{→}_{1≤i,j≤r}^{(τ)}(V_{ij}) as in (II.15) we see that the same cancellation argument

applies. Thus we have det^{→}_{1≤i,j≤r}^{(τ)}(V_{ij}) = det^{→}_{1≤i,j≤r}(V_{ij}), which proves (II.14).

Remark II.17. Note that in the determinant of Proposition II.9, we may not expand along rows because then the cancellation argument does not apply. (For a counterexample, consider the r = 2 case.)

Corollary II.18. Let W_{i}, V_{ij} be linear operators acting on formal Laurent series, c_{j}(k) arbitrary constants for k ∈ ℤ^{r} and i, j = 1, …, r. Suppose V_{ij} = C_{ij} + A_{ij}, with the operators C_{ij}, A_{ij}, i, j = 1, …, r, satisfying the conditions (II.10), (II.11), and (II.12) of Proposition II.9. Suppose W_{i} = W_{i}^{(c)} + W_{i}^{(a)}, with the operators W_{i}, W_{i}^{(c)}, W_{i}^{(a)}, i = 1, …, r, satisfying

C_{kl}W_{i} = W_{i}C_{kl},  i ≠ k; i, k, l = 1, …, r,  (II.19)
A_{kl}W_{i}^{(c)} = W_{i}^{(c)}A_{kl},  i ≠ k; i, k, l = 1, …, r,  (II.20)
A_{kl}W_{i}^{(a)} = A_{il}W_{k}^{(a)},  i, k, l = 1, …, r.  (II.21)

Moreover the c_{j}(k) are assumed to satisfy (II.4), and det^{→}_{1≤i,j≤r}(V_{ij}) is assumed to be invertible. With the notation of Lemma II.2, if

∑_{j=1}^{r} c_{j}(k)V_{ij}f_{k}(z) = W_{i}f_{k}(z),  i = 1, …, r,  (II.22)

then

g_{k}(z) = \frac{1}{⟨f_{k}(z), det^{→}(V_{ij}^{∗})h_{k}(z)⟩} det^{→}(V_{ij}^{∗})h_{k}(z),  (II.23)

where h_{k}(z) is a solution of

∑_{j=1}^{r} c_{j}(k)V_{ij}^{∗}h_{k}(z) = W_{i}^{∗}h_{k}(z),  i = 1, …, r,  (II.24)

with h_{k}(z) ≢ 0 for all k ∈ ℤ^{r}.

Proof. By Proposition II.9 it follows that the column determinant det^{→}(V_{ij}) may be expanded along any column. Hence, we can apply Cramer’s rule to (II.22) to obtain

c_{j}(k) det^{→}_{1≤i,l≤r}(V_{il})f_{k}(z) = ∑_{i=1}^{r}(−1)^{i+j}V⃗^{(i,j)}W_{i}f_{k}(z),

for j = 1, …, r. The dual system reads

c_{j}(k) det^{→}_{1≤i,l≤r}(V_{il}^{∗})h_{k}(z) = ∑_{i=1}^{r}(−1)^{i+j}W_{i}^{∗}V⃗^{∗}^{(i,j)}h_{k}(z)  (II.25)

= ∑_{i=1}^{r}(−1)^{i+j}V⃗^{∗}^{(i,j)}W_{i}^{∗}h_{k}(z),

for j = 1, …, r, and is equivalent to (II.24). Notice that, because of (II.19), (II.20), and (II.21), we may apply Proposition II.9 in (II.25) and shift W_{i}^{∗} to the right. Now apply Lemma II.2 with V = det^{→}(V_{ij}) and U_{j} = ∑_{i=1}^{r}(−1)^{i+j}V⃗^{(i,j)}W_{i}.

16 II. AN OPERATOR METHOD FOR PROVING MATRIX INVERSIONS

CHAPTER III Multidimensional matrix inversions

In this chapter, we utilize the operator method of chapter II to derive two multidimensional matrix inversions, Theorem III.7 and Theorem III.29. We use special cases of these inversions to derive new multiple basic hypergeometric summation formulas in chapter IV. The matrix inversion in Theorem III.7 is an r-dimensional extension of a matrix inversion recently found by Krattenthaler [45] which contains a host of previously known one-dimensional matrix inversions. Our r-dimensional extension thereof generalizes multidimensional matrix inversions recently found by Milne, Lilly and Milne, and Bhatnagar and Milne, respectively, see Remark III.10. Our second matrix inversion in Theorem III.29 is of different type although it also includes the multidimensional matrix inversion of Lilly and Milne as a special case, see Remark III.32.

To motivate our multidimensional work in this chapter we recall Krattenthaler’s matrix inverse with its proof [45] using the r = 1 case of Lemma II.2.

Theorem III.1 (Krattenthaler’s matrix inverse). Let (a_{i})_{i∈ℤ}, (c_{i})_{i∈ℤ} be arbitrary sequences such that c_{i} ≠ c_{j} if i ≠ j. Then (f_{nk})_{n,k∈ℤ} and (g_{kl})_{k,l∈ℤ} are inverses of each other, where

f_{nk} = \frac{∏_{j=k}^{n−1}(a_{j} − c_{k})}{∏_{j=k+1}^{n}(c_{j} − c_{k})},  (III.2)

and

g_{kl} = \frac{(a_{l} − c_{l})}{(a_{k} − c_{k})} \frac{∏_{j=l+1}^{k}(a_{j} − c_{k})}{∏_{j=l}^{k−1}(c_{j} − c_{k})}.  (III.3)

Proof. From (III.2) we deduce for n ≥ k the recursion

(c_{n} − c_{k})f_{nk} = (a_{n−1} − c_{k})f_{n−1,k}.  (III.4)

We write

f_{k}(z) = ∑_{n≥k} \frac{∏_{j=k}^{n−1}(a_{j} − c_{k})}{∏_{j=k+1}^{n}(c_{j} − c_{k})} z^{n}.

Moreover, we define linear operators A, C by Az^{n} = a_{n}z^{n} and Cz^{n} = c_{n}z^{n}. Then we may write (III.4) in the form

(C − zA)f_{k}(z) = c_{k}(1 − z)f_{k}(z),  (III.5)

valid for all k ∈ ℤ. Now we may apply the r = 1 case of Lemma II.2, since (III.5) is a system of type (II.3) with U = C − zA, V = 1 − z and the functions c_{k} satisfy (II.4). The dual system (II.5) for the auxiliary formal Laurent series h_{k}(z) in this case reads

(C^{∗} − A^{∗}z)h_{k}(z) = c_{k}(1 − z)h_{k}(z),  (III.6)

As is easily seen, we have A^{∗}z^{−l} = a_{l}z^{−l} and C^{∗}z^{−l} = c_{l}z^{−l}. Thus, with h_{k}(z) = ∑_{l≤k} h_{kl}z^{−l}, by comparing coefficients of z^{−l} in (III.6) we obtain

(c_{l} − c_{k})h_{kl} = (a_{l} − c_{k})h_{k,l+1}.

If we set h_{kk} = 1, we get

h_{kl} = \frac{∏_{j=l}^{k−1}(a_{j} − c_{k})}{∏_{j=l}^{k−1}(c_{j} − c_{k})}.

Taking into account (II.6), here we have g_{k}(z) = (1 − z)h_{k}(z), where g_{k}(z) = ∑_{l≤k} g_{kl}z^{−l}. (Since f_{kk} = 1, the pairing ⟨f_{k}(z), V^{∗}h_{k}(z)⟩ is simply the coefficient of z^{−k} in (1 − z)h_{k}(z) and equals 1.) So, comparing coefficients of z^{−l} leads to

g_{kl} = h_{kl} − h_{k,l+1} = \frac{(a_{l} − c_{l})}{(a_{k} − c_{k})} \frac{∏_{j=l+1}^{k}(a_{j} − c_{k})}{∏_{j=l}^{k−1}(c_{j} − c_{k})},

which is exactly (III.3).

Now we present one of the main results of this thesis, a multidimensional extension of Theorem III.1. For convenience, we introduce the notation |n| = n_{1} + n_{2} + ⋯ + n_{r}.

Theorem III.7. Let (a_{t})_{t∈ℤ}, (c_{i}(t_{i}))_{t_{i}∈ℤ}, i = 1, …, r be arbitrary sequences, b arbitrary, such that none of the denominators in (III.8) or (III.9) vanish. Then (f_{nk})_{n,k∈ℤ^{r}} and (g_{kl})_{k,l∈ℤ^{r}} are inverses of each other, where

f_{nk} = \frac{∏_{t=|k|}^{|n|−1}(a_{t} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{i=1}^{r} ∏_{t_{i}=k_{i}+1}^{n_{i}}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} \frac{∏_{i=1}^{r} ∏_{t=|k|}^{|n|−1}(a_{t} − c_{i}(k_{i}))}{∏_{i,j=1}^{r} ∏_{t_{i}=k_{i}+1}^{n_{i}}(c_{i}(t_{i}) − c_{j}(k_{j}))} (III.8)

and

g_{kl} = ∏_{1≤i<j≤r} \frac{(c_{i}(l_{i}) − c_{j}(l_{j}))}{(c_{i}(k_{i}) − c_{j}(k_{j}))}

× \frac{(b − a_{|l|} ∏_{j=1}^{r} c_{j}(l_{j}))}{(b − a_{|k|} ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i=1}^{r} \frac{(a_{|l|} − c_{i}(l_{i}))}{(a_{|k|} − c_{i}(k_{i}))}

× \frac{∏_{t=|l|+1}^{|k|}(a_{t} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{i=1}^{r} ∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} \frac{∏_{i=1}^{r} ∏_{t=|l|+1}^{|k|}(a_{t} − c_{i}(k_{i}))}{∏_{i,j=1}^{r} ∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − c_{j}(k_{j}))}. (III.9)

Remark III.10. The special case a_{t} = 0, c_{j}(k_{j}) = x_{j}^{−1}q^{−k_{j}} is equivalent to the A_{r} Bailey transform of Milne and Lilly [55], [57], the specialization a_{t} = 0, c_{j}(k_{j}) = x_{j}^{−1}q^{−k_{j}} +x_{j}q^{k_{j}}, b = 0 is equivalent to their C_{r} Bailey transform [57], [47], [48]. The limiting case a_{t} = b aq^{−t}, c_{j}(k_{j}) = x_{j}^{−1}q^{−k_{j}}, then b → 0, is equivalent to a second A_{r} Bailey transform of Milne [55, Theorem 8.26]. The specialization c_{j}(k_{j}) = x_{j}^{−1}q^{−k_{j}} is equivalent to the A_{r} matrix inverse of Bhatnagar and Milne [11, Theorem 5.7], [13, Theorem 3.48]. Moreover, the r = 1 case is a restatement of Krattenthaler’s matrix inversion, Theorem III.1. Due to the fact that Theorem III.7 covers all known A_{r} matrix inversions (to the author’s knowledge), we view Theorem III.7 as an A_{r} matrix inversion theorem (also see Remark III.32).

Another important special case of Theorem III.7 is a new multidimensional bibasic hypergeometric matrix inversion, stated separately as Corollary III.25, which we utilize in our applications in chapter IV.

Proof of Theorem III.7. We will use the operator method of chapter II. From (III.8) we deduce for n ≥ k the recursion

(c_{i}(n_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(c_{i}(n_{i}) − c_{s}(k_{s}))f_{nk}

= (a_{|n|−1} − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(a_{|n|−1} − c_{s}(k_{s}))f_{n−e_{i},k}, (III.11)

for i = 1, …, r, where e_{i} denotes the vector of ℤ^{r} where all components are zero except the i-th, which is 1. We write

f_{k}(z) = ∑_{n≥k} \frac{∏_{t=|k|}^{|n|−1}(a_{t} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{i=1}^{r} ∏_{t_{i}=k_{i}+1}^{n_{i}}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} \frac{∏_{i=1}^{r} ∏_{t=|k|}^{|n|−1}(a_{t} − c_{i}(k_{i}))}{∏_{i,j=1}^{r} ∏_{t_{i}=k_{i}+1}^{n_{i}}(c_{i}(t_{i}) − c_{j}(k_{j}))}z^{n}.

Moreover, we define linear operators A, C_{i} by Az^{n} = a_{|n|}z^{n} and C_{i}z^{n} = c_{i}(n_{i})z^{n} for all i = 1, …, r. Then we may write (III.11) in the form

(C_{i} − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(C_{i} − c_{s}(k_{s}))f_{k}(z)

= z_{i}(A − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(A − c_{s}(k_{s}))f_{k}(z), (III.12)

valid for all k ∈ ℤ^{r}. We want to write our system of equations in a way such that Corollary II.18 is applicable. In order to achieve this, we expand the products on both sides of (III.12) in terms of the elementary symmetric functions (see [49, p.19])

e_{j}(c_{1}(k_{1}), c_{2}(k_{2}), …, c_{r}(k_{r}), b/ ∏_{s=1}^{r} c_{s}(k_{s}))

of order j, for which we write e_{j}(c(k)) for short. Our recurrence system then reads, using e_{r+1}(c(k)) = b,

∑_{j=1}^{r} e_{j}(c(k))[(−C_{i})^{r+1−j} − z_{i}(−A)^{r+1−j}]f_{k}(z)

= [z_{i}(−A)^{r+1} + bz_{i} − (−C_{i})^{r+1} − b]f_{k}(z), i = 1, …, r. (III.13)

Now (III.13) is a system of type (II.22) with V_{ij} = [(−C_{i})^{r+1−j} − z_{i}(−A)^{r+1−j}], W_{i} = [z_{i}(−A)^{r+1} + bz_{i} − (−C_{i})^{r+1} − b], and c_{j}(k) = e_{j}(c(k)). The operators C_{ij} = (−C_{i})^{r+1−j}, A_{ij} = −z_{i}(−A)^{r+1−j}, W_{i}^{(c)} = [−(−C_{i})^{r+1} −b], W_{i}^{(a)} = [z_{i}(−A)^{r+1} +bz_{i}] satisfy (II.10), (II.11), (II.12), (II.19), (II.20), and (II.21), the functions c_{j}(k) satisfy (II.4). Hence we may apply Corollary II.18. The dual system (II.24) for the auxiliary formal Laurent series h_{k}(z) in this case reads

∑_{j=1}^{r} e_{j}(c(k))[(−C_{i}^{∗})^{r+1−j} − (−A^{∗})^{r+1−j}z_{i}]h_{k}(z)

= [(−A^{∗})^{r+1}z_{i} + bz_{i} − (−C_{i}^{∗})^{r+1} − b]h_{k}(z), i = 1, …, r.

Equivalently, we have

(C_{i}^{∗} − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(C_{i}^{∗} − c_{s}(k_{s}))h_{k}(z)

= (A^{∗} − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(A^{∗} − c_{s}(k_{s}))z_{i}h_{k}(z), (III.14)

for all i = 1, …, r and k ∈ ℤ^{r}. As is easily seen, we have A^{∗}z^{−l} = a_{|l|}z^{−l} and C_{i}^{∗}z^{−l} = c_{i}(l_{i})z^{−l} for i = 1, …, r. Thus, with h_{k}(z) = ∑_{l≤k} h_{kl}z^{−l}, by comparing coefficients of z^{−l} in (III.14) we obtain

(c_{i}(l_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(c_{i}(l_{i}) − c_{s}(k_{s}))h_{kl}

= (a_{|l|} − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(a_{|l|} − c_{s}(k_{s}))h_{k,l+e_{i}}.

If we set h_{kk} = 1, we get

h_{kl} = \frac{∏_{t=|l|}^{|k|−1}(a_{t} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{i=1}^{r} ∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} \frac{∏_{i=1}^{r} ∏_{t=|l|}^{|k|−1}(a_{t} − c_{i}(k_{i}))}{∏_{i,j=1}^{r} ∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − c_{j}(k_{j}))}.

Taking into account (II.23), we have to compute the action of

det^{→}_{1≤i,j≤r}(V_{ij}^{∗}) = det^{→}_{1≤i,j≤r}[(−C_{i}^{∗})^{r+1−j} − (−A^{∗})^{r+1−j}z_{i}]  (III.15)

when applied to

h_{k}(z) = ∑_{l≤k} \frac{∏_{t=|l|}^{|k|−1}(a_{t} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{i=1}^{r} ∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} \frac{∏_{i=1}^{r} ∏_{t=|l|}^{|k|−1}(a_{t} − c_{i}(k_{i}))}{∏_{i,j=1}^{r} ∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − c_{j}(k_{j}))}z^{−l}.

From Proposition II.9 it follows that the determinant (III.15) can be written as

det^{→}_{1≤i,j≤r}(V_{ij}^{∗}) = det^{→}_{1≤i,j≤r}(C_{ij}^{∗}) + ∑_{i,j=1}^{r}(−1)^{i+j} det^{→}_{1≤m≤r,m≠i}_{1≤n≤r,n≠j}(C_{mn}^{∗}) A_{ij}^{∗},

or more explicitly,

det^{→}_{1≤i,j≤r}(V_{ij}^{∗}) = det^{→}_{1≤i,j≤r}[(−C_{i}^{∗})^{r+1−j} − (−A^{∗})^{r+1−j}z_{i}]

= det^{→}_{1≤i,j≤r}[(−C_{i}^{∗})^{r+1−j}]

+ ∑_{i,j=1}^{r}(−1)^{i+j} det^{→}_{1≤m≤r,m≠i}_{1≤n≤r,n≠j}[(−C_{m}^{∗})^{r+1−n}] (−(−A^{∗})^{r+1−j}z_{i}) . (III.16)

Note that after expanding the column determinants by (II.16) all summands in (III.16) have pairwise commuting factors (when regarding (−(−A^{∗})^{r+1−j}z_{i}) as one factor). Since

z_{i}h_{k}(z) = ∑_{l≤k} \frac{(c_{i}(l_{i}) − b/ ∏_{s=1}^{r} c_{s}(k_{s}))}{(a_{|l|} − b/ ∏_{s=1}^{r} c_{s}(k_{s}))} ∏_{s=1}^{r} \frac{(c_{i}(l_{i}) − c_{s}(k_{s}))}{(a_{|l|} − c_{s}(k_{s}))}h_{kl}z^{−l},

we conclude that

det^{→}_{1≤i,j≤r}(V_{ij}^{∗})h_{k}(z)

= ∑_{l≤k} ( det^{→}_{1≤i,j≤r}(C_{ij}^{∗}) + ∑_{i,j=1}^{r}(−1)^{i+j} det^{→}_{1≤m≤r,m≠i}_{1≤n≤r,n≠j}(C_{mn}^{∗}) A_{ij}^{∗})h_{kl}z^{−l}

= ∑_{l≤k} ( det^{→}_{1≤i,j≤r} [(−c_{i}(l_{i}))^{r+1−j}]

+ ∑_{i,j=1}^{r}(−1)^{i+j} det^{→}_{1≤m≤r,m≠i}_{1≤n≤r,n≠j} [(−c_{m}(l_{m}))^{r+1−n}] (−(−a_{|l|})^{r+1−j})

· \frac{(c_{i}(l_{i}) − b/ ∏_{s=1}^{r} c_{s}(k_{s}))}{(a_{|l|} − b/ ∏_{s=1}^{r} c_{s}(k_{s}))} ∏_{s=1}^{r} \frac{(c_{i}(l_{i}) − c_{s}(k_{s}))}{(a_{|l|} − c_{s}(k_{s}))})h_{kl}z^{−l}. (III.17)

We claim that

det^{→}_{1≤i,j≤r}(V_{ij}^{∗})h_{k}(z) = ∑_{l≤k} det_{1≤i,j≤r}(v_{ij})h_{kl}z^{−l},  (III.18)

where

v_{ij} = (−c_{i}(l_{i}))^{r+1−j} − (−a_{|l|})^{r+1−j} \frac{(c_{i}(l_{i}) − b/ ∏_{s=1}^{r} c_{s}(k_{s}))}{(a_{|l|} − b/ ∏_{s=1}^{r} c_{s}(k_{s}))} ∏_{s=1}^{r} \frac{(c_{i}(l_{i}) − c_{s}(k_{s}))}{(a_{|l|} − c_{s}(k_{s}))}.

The claim follows from the observation that det_{1≤i,j≤r}(v_{ij}) is a determinant of commuting entries and so trivially satisfies the assumptions of Proposition II.9. Thus ∑_{l≤k} det_{1≤i,j≤r}(v_{ij})h_{kl}z^{−l} can also be transformed into the right side of (III.17).

For the computation of det_{1≤i,j≤r}(v_{ij}) we utilize Lemma A.1 with x_{i} = −c_{i}(l_{i}), y_{s} = −c_{s}(k_{s}), a = −a_{|l|}, and c = (−1)^{r+1}b, obtaining

det_{1≤i,j≤r}(v_{ij}) =\frac{(a_{|l|} − b/ ∏_{j=1}^{r} c_{j}(l_{j}))}{(a_{|l|} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i=1}^{r} \frac{(a_{|l|} − c_{i}(l_{i}))}{(a_{|l|} − c_{i}(k_{i}))}

× ∏_{i=1}^{r}(−c_{i}(l_{i})) ∏_{1≤i<j≤r}(c_{j}(l_{j}) − c_{i}(l_{i})).

Plugging this determinant evaluation into (III.18) leads to

det^{→}_{1≤i,j≤r}(V_{ij}^{∗})h_{k}(z) = ∑_{l≤k} ( ∏_{1≤i<j≤r}(c_{j}(l_{j}) − c_{i}(l_{i})) ∏_{i=1}^{r}(−c_{i}(l_{i}))

× \frac{(a_{|l|} − b/ ∏_{j=1}^{r} c_{j}(l_{j}))}{(a_{|k|} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i=1}^{r} \frac{(a_{|l|} − c_{i}(l_{i}))}{(a_{|k|} − c_{i}(k_{i}))}

× \frac{∏_{t=|l|+1}^{|k|}(a_{t} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{i=1}^{r} ∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} \frac{∏_{i=1}^{r} ∏_{t=|l|+1}^{|k|}(a_{t} − c_{i}(k_{i}))}{∏_{i,j=1}^{r} ∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − c_{j}(k_{j}))}z^{−l}). (III.19)

Note that since f_{kk} = 1, the pairing ⟨f_{k}(z), det^{→}(V_{ij}^{∗})h_{k}(z)⟩ is simply the coefficient of z^{−k} in (III.19). Thus, equation (II.23) reads

g_{k}(z) = ∏_{1≤i<j≤r}(c_{j}(k_{j}) − c_{i}(k_{i}))^{−1} ∏_{i=1}^{r}(−c_{i}(k_{i}))^{−1} det^{→}_{1≤i,j≤r}(V_{ij}^{∗})h_{k}(z), (III.20)

where g_{k}(z) = ∑_{l≤k} g_{kl}z^{−l}. So, extracting the coefficient of z^{−l} in (III.20) we obtain exactly (III.9).

For reference, we give two important basic hypergeometric specializations of Theorem III.7.

Corollary III.21 (Bhatnagar and Milne). Let

f_{nk} = \frac{(ap^{|k|}q^{−|k|}/d; p)_{|n|−|k|}}{∏_{i=1}^{r}(dx_{i}q^{1+k_{i}+|k|}; q)_{n_{i}−k_{i}}} \frac{∏_{i=1}^{r}(ax_{i}p^{|k|}q^{k_{i}}; p)_{|n|−|k|}}{∏_{i,j=1}^{r}(q^{1+k_{i}−k_{j}}x_{i}/x_{j}; q)_{n_{i}−k_{i}}}  (III.22)

and

g_{kl} = (−1)^{|k|−|l|} q^{\binom{|k|−|l|}{2}} \frac{(1 − ap^{|l|}q^{−|l|}/d)}{(1 − ap^{|k|}q^{−|k|})} ∏_{i=1}^{r} \frac{(1 − ax_{i}p^{|l|}q^{l_{i}})}{(1 − ax_{i}p^{|k|}q^{k_{i}})}

× \frac{(ap^{1+|l|}q^{−|k|}/d; p)_{|k|−|l|}}{∏_{i=1}^{r}(dx_{i}q^{l_{i}+|k|}; q)_{k_{i}−l_{i}}} \frac{∏_{i=1}^{r}(ax_{i}p^{1+|l|}q^{k_{i}}; p)_{|k|−|l|}}{∏_{i,j=1}^{r}(q^{1+l_{i}−l_{j}}x_{i}/x_{j}; q)_{k_{i}−l_{i}}} . (III.23)

Then (f_{nk})_{n,k∈ℤ^{r}} and (g_{kl})_{k,l∈ℤ^{r}} are infinite lower-triangular r-dimensional matrices being inverses of each other.

Proof. In Theorem III.7 we set a_{t} ↦ aq^{t}, b ↦ d/x_{1} ⋯ x_{n}, and c_{i}(t_{i}) ↦ q^{−t_{i}}/x_{i} for i = 1, …, r. After some elementary manipulations which include Lemma B.1 we obtain the inverse pair (III.22) and (III.23).

Remark III.24. The matrix inversion in Corollary III.21 is an A_{r} extension of Gasper’s bibasic matrix inversion [24, (3.2) and (3.3)], [25, (3.6.19) and (3.6.20)], to which it reduces for r = 1. This inversion can easily be obtained from the inversion [13, Theorem 3.48] of Bhatnagar and Milne.

Corollary III.25. Let

f_{nk} = \begin{matrix}∏_{i=1}^{r} & [(ap^{|k|}q^{k_{i}}x_{i}; p)_{|n|−|k|} (ap^{|k|}q^{−k_{i}}/x_{i}; p)_{|n|−|k|}] \\ ∏_{i,j=1}^{r} & [(q^{1+k_{i}−k_{j}}x_{i}/x_{j}; q)_{n_{i}−k_{i}} (q^{1+k_{i}+k_{j}}x_{i}x_{j}; q)_{n_{i}−k_{i}}] \end{matrix}  (III.26)

and

g_{kl} = (−1)^{|k|−|l|} q^{\binom{|k|−|l|}{2}} ∏_{1≤i<j≤r} \frac{(1 − x_{i}x_{j}q^{l_{i}+l_{j}})}{(1 − x_{i}x_{j}q^{k_{i}+k_{j}})}

× ∏_{i=1}^{r} \frac{(1 − ap^{|l|}q^{l_{i}}x_{i})(1 − ap^{|l|}q^{−l_{i}}/x_{i})}{(1 − ap^{|k|}q^{k_{i}}x_{i})(1 − ap^{|k|}q^{−k_{i}}/x_{i})}

× \begin{matrix}∏_{i=1}^{r} & [(ap^{1+|l|}q^{k_{i}}x_{i}; p)_{|k|−|l|} (ap^{1+|l|}q^{−k_{i}}/x_{i}; p)_{|k|−|l|}] \\ ∏_{i,j=1}^{r} & [(q^{1+l_{i}−l_{j}}x_{i}/x_{j}; q)_{k_{i}−l_{i}} (q^{l_{i}+k_{j}}x_{i}x_{j}; q)_{k_{i}−l_{i}}] \end{matrix}. (III.27)

Then (f_{nk})_{n,k∈ℤ^{r}} and (g_{kl})_{k,l∈ℤ^{r}} are infinite lower-triangular r-dimensional matrices being inverses of each other.

Proof. In Theorem III.7 we replace b ↦ 0, a_{t} ↦ ap^{t} + p^{−t}/a, and c_{i}(t_{i}) ↦ x_{i}q^{t_{i}} + q^{−t_{i}}/x_{i} for i = 1, …, r. After some elementary manipulations, including Lemma B.1, we obtain the inverse pair (III.26) and (III.27).

Remark III.28. The inversion in Corollary III.25 is a D_{r} extension of Gasper’s bibasic matrix inversion [24, (3.2) and (3.3)], [25, (3.6.19) and (3.6.20)], to which it reduces for r = 1.

Next, we give another multidimensional matrix inversion which we can prove by the operator method of chapter II. The multidimensional matrix inversion in Theorem III.29 is of different type than the inversion of Theorem III.7, see Remark III.32.

Theorem III.29. Let (c_{i}(t_{i}))_{t_{i}∈ℤ}, i = 1, …, r, be arbitrary sequences, b arbitrary, such that none of the denominators in (III.30) or (III.31) vanish. Then (f_{nk})_{n,k∈ℤ^{r}} and (g_{kl})_{k,l∈ℤ^{r}} are inverses of each other, where

f_{nk} = ∏_{i=1}^{r} \frac{∏_{t_{i}=k_{i}}^{n_{i}−1}(1 − bc_{i}(t_{i})/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{t_{i}=k_{i}+1}^{n_{i}}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i,j=1}^{r} \frac{∏_{t_{i}=k_{i}}^{n_{i}−1}(1 − c_{i}(t_{i})c_{j}(k_{j}))}{∏_{t_{i}=k_{i}+1}^{n_{i}}(c_{i}(t_{i}) − c_{j}(k_{j}))} (III.30)

and

g_{kl} = ∏_{1≤i<j≤r} (\frac{(c_{i}(l_{i}) − c_{j}(l_{j}))}{(c_{i}(k_{i}) − c_{j}(k_{j}))} \frac{(1 − c_{i}(l_{i})c_{j}(l_{j}))}{(1 − c_{i}(k_{i})c_{j}(k_{j}))})

× ∏_{i=1}^{r} \frac{(1 − c_{i}(l_{i})^{2})}{(1 − c_{i}(k_{i})^{2})} ∏_{i=1}^{r} \frac{c_{i}(l_{i})}{c_{i}(k_{i})}

× ∏_{i=1}^{r} \frac{∏_{t_{i}=l_{i}+1}^{k_{i}}(1 − bc_{i}(t_{i})/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i,j=1}^{r} \frac{∏_{t_{i}=l_{i}+1}^{k_{i}}(1 − c_{i}(t_{i})c_{j}(k_{j}))}{∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − c_{j}(k_{j}))}. (III.31)

Remark III.32. The special case c_{j}(k_{j}) = x_{j}^{−1}q^{−k_{j}} is a C_{r} generalization of Bressoud’s matrix inversion formula [16], as pointed out in [48, second remark after Theorem 2.11]. Setting, in addition, b = 0 yields a C_{r} Bailey transform which is equivalent to the one derived in [48]. Therefore, we view Theorem III.29 as a C_{r} matrix inversion theorem. See Corollary III.40 for a C_{r} Bressoud’s matrix inverse.

Proof of Theorem III.29. Again, we will use the operator method of chapter II. From (III.30) we deduce for n ≥ k the recursion

(c_{i}(n_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(c_{i}(n_{i}) − c_{s}(k_{s}))f_{nk}

= (1 − bc_{i}(n_{i} − 1)/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(1 − c_{i}(n_{i} − 1)c_{s}(k_{s}))f_{n−e_{i},k}, (III.33)

for i = 1, …, r. We write

f_{k}(z) = ∑_{l≤k} ∏_{i=1}^{r} \frac{∏_{t_{i}=k_{i}}^{n_{i}−1}(1 − bc_{i}(t_{i})/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{t_{i}=k_{i}+1}^{n_{i}}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i,j=1}^{r} \frac{∏_{t_{i}=k_{i}}^{n_{i}−1}(1 − c_{i}(t_{i})c_{j}(k_{j}))}{∏_{t_{i}=k_{i}+1}^{n_{i}}(c_{i}(t_{i}) − c_{j}(k_{j}))}z^{n}.

Moreover, we define linear operators C_{i} by C_{i}z^{n} = c_{i}(n_{i})z^{n} for i = 1, …, r. Then we may write (III.33) in the form

(C_{i} − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(C_{i} − c_{s}(k_{s}))f_{k}(z)

= z_{i}(I − C_{i}b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(I − C_{i}c_{s}(k_{s}))f_{k}(z), (III.34)

I being the identity operator, valid for all k ∈ ℤ^{r}. We will write our system of equations in a way such that Corollary II.18 is applicable. Again, we expand the products on both sides of (III.34) in terms of the elementary symmetric functions

e_{j}(c_{1}(k_{1}), c_{2}(k_{2}), …, c_{r}(k_{r}), b/ ∏_{s=1}^{r} c_{s}(k_{s}))

of order j, for which we write e_{j}(c(k)) for short. Our recurrence system then reads, again using e_{r+1}(c(k)) = b,

∑_{j=1}^{r} e_{j}(c(k))[(−C_{i})^{r+1−j} − (−1)^{r+1}z_{i}(−C_{i})^{j}]f_{k}(z)

= [(−1)^{r+1}z_{i} + bz_{i}C_{i}^{r+1} − (−C_{i})^{r+1} − b]f_{k}(z), i = 1, …, r. (III.35)

Now (III.35) is a system of type (II.22) with V_{ij} = [(−C_{i})^{r+1−j} − (−1)^{r+1}z_{i}(−C_{i})^{j}], W_{i} = [(−1)^{r+1}z_{i}+bz_{i}C_{i}^{r+1}−(−C_{i})^{r+1}−b] and c_{j}(k) = e_{j}(c(k)). The operators C_{ij} = V_{ij}, A_{ij} = 0, W_{i}^{(c)} = W_{i}, W_{i}^{(a)} = 0 satisfy (II.10), (II.11), (II.12), (II.19), (II.20), and (II.21), the functions c_{j}(k) satisfy (II.4). Hence we may apply Corollary II.18. The dual system (II.24) for the auxiliary formal Laurent series h_{k}(z) in this case reads

∑_{j=1}^{r} e_{j}(c(k))[(−C_{i}^{∗})^{r+1−j} − (−1)^{r+1}(−C_{i}^{∗})^{j}z_{i}]h_{k}(z)

= [(−1)^{r+1}z_{i} + b(C_{i}^{∗})^{r+1}z_{i} − (−C_{i}^{∗})^{r+1} − b]h_{k}(z), i = 1, …, r.

III. MULTIDIMENSIONAL MATRIX INVERSIONS 27

Equivalently, we have

(C_{i}^{∗} − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(C_{i}^{∗} − c_{s}(k_{s}))h_{k}(z)

= (I − C_{i}^{∗}b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(I − C_{i}^{∗}c_{s}(k_{s}))z_{i}h_{k}(z), (III.36)

for all i = 1, …, r and k ∈ ℤ^{r}. Thus, with h_{k}(z) = ∑_{l≤k} h_{kl}z^{−l}, by comparing coefficients of z^{−l} in (III.36) we obtain

(c_{i}(l_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(c_{i}(l_{i}) − c_{s}(k_{s}))h_{kl}

= (1 − bc_{i}(l_{i})/ ∏_{j=1}^{r} c_{j}(k_{j})) ∏_{s=1}^{r}(1 − c_{i}(l_{i})c_{s}(k_{s}))h_{k,l+e_{i}}.

If we set h_{kk} = 1, we get

h_{kl} = ∏_{i=1}^{r} \frac{∏_{t_{i}=l_{i}}^{k_{i}−1}(1 − bc_{i}(t_{i})/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i,j=1}^{r} \frac{∏_{t_{i}=l_{i}}^{k_{i}−1}(1 − c_{i}(t_{i})c_{j}(k_{j}))}{∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − c_{j}(k_{j}))}.

Taking into account (II.23), we have to compute the action of

det^{→}_{1≤i,j≤r}(V_{ij}^{∗}) = det^{→}_{1≤i,j≤r}[(−C_{i}^{∗})^{r+1−j} − (−1)^{r+1}(−C_{i}^{∗})^{j}z_{i}]

when applied to

h_{k}(z) = ∑_{l≤k} ∏_{i=1}^{r} \frac{∏_{t_{i}=l_{i}}^{k_{i}−1}(1 − bc_{i}(t_{i})/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i,j=1}^{r} \frac{∏_{t_{i}=l_{i}}^{k_{i}−1}(1 − c_{i}(t_{i})c_{j}(k_{j}))}{∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − c_{j}(k_{j}))}z^{−l}.

Because V_{i_{1}j_{1}}^{∗} and V_{i_{2}j_{2}}^{∗} commute for i_{1} ≠ i_{2} and all j_{1}, j_{2}, all the summands in det^{→}_{1≤i,j≤r}(V_{ij}^{∗}) have pairwise commuting factors. Since

z_{i}h_{k}(z) = ∑_{l≤k} \frac{(c_{i}(l_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))}{(1 − bc_{i}(l_{i})/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{j=1}^{r} \frac{(c_{i}(l_{i}) − c_{j}(k_{j}))}{(1 − c_{i}(l_{i})c_{j}(k_{j}))}h_{kl}z^{−l},

we conclude that

det^{→}_{1≤i,j≤r}(V_{ij}^{∗})h_{k}(z) = ∑_{l≤k} det_{1≤i,j≤r}(v_{ij})h_{kl}z^{−l},  (III.37)

where

v_{ij} =(−c_{i}(l_{i}))^{r+1−j}

− (−1)^{r+1}(−c_{i}(l_{i}))^{j} \frac{(c_{i}(l_{i}) − b/ ∏_{s=1}^{r} c_{s}(k_{s}))}{(1 − bc_{i}(l_{i})/ ∏_{s=1}^{r} c_{s}(k_{s}))} ∏_{s=1}^{r} \frac{(c_{i}(l_{i}) − c_{s}(k_{s}))}{(1 − c_{i}(l_{i})c_{s}(k_{s}))}.

For the computation of det_{1≤i,j≤r}(v_{ij}) we utilize Lemma A.11 with x_{i} = −c_{i}(l_{i}), y_{s} = −c_{s}(k_{s}), and c = (−1)^{r−1}b, obtaining

det_{1≤i,j≤r}(v_{ij}) = ∏_{i=1}^{r} \frac{(1 − c_{i}(k_{i})b/ ∏_{j=1}^{r} c_{j}(k_{j}))}{(1 − c_{i}(l_{i})b/ ∏_{j=1}^{r} c_{j}(k_{j}))}

× ∏_{i=1}^{r}(1 − c_{i}(l_{i})^{2}) ∏_{i=1}^{r}(−c_{i}(l_{i})) ∏_{i,j=1}^{r}(1 − c_{i}(l_{i})c_{j}(k_{j}))^{−1}

× ∏_{1≤i<j≤r}[(c_{j}(l_{j}) − c_{i}(l_{i}))(1 − c_{i}(l_{i})c_{j}(l_{j}))(1 − c_{i}(k_{i})c_{j}(k_{j}))].

Plugging this determinant evaluation into (III.37) leads to

det^{→}_{1≤i,j≤r}(V_{ij}^{∗})h_{k}(z) = ∑_{l≤k} ( ∏_{1≤i<j≤r}[(c_{j}(l_{j}) − c_{i}(l_{i}))(1 − c_{i}(l_{i})c_{j}(l_{j}))]

× ∏_{i=1}^{r}(1 − c_{i}(l_{i})^{2}) ∏_{i=1}^{r}(−c_{i}(l_{i})) ∏_{i≤j}(1 − c_{i}(k_{i})c_{j}(k_{j}))^{−1}

× ∏_{i=1}^{r} \frac{∏_{t_{i}=l_{i}+1}^{k_{i}}(1 − bc_{i}(t_{i})/ ∏_{j=1}^{r} c_{j}(k_{j}))}{∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{i,j=1}^{r} \frac{∏_{t_{i}=l_{i}+1}^{k_{i}}(1 − c_{i}(t_{i})c_{j}(k_{j}))}{∏_{t_{i}=l_{i}}^{k_{i}−1}(c_{i}(t_{i}) − c_{j}(k_{j}))}z^{−l}). (III.38)

Again, the pairing ⟨f_{k}(z), det^{→}(V_{ij}^{∗})h_{k}(z)⟩ is simply the coefficient of z^{−k} in (III.38). Thus, equation (II.23) reads

g_{k}(z) = ∏_{1≤i<j≤r}(c_{j}(k_{j}) − c_{i}(k_{i}))^{−1} ∏_{i=1}^{r}(−c_{i}(k_{i}))^{−1} det^{→}_{1≤i,j≤r}(V_{ij}^{∗})h_{k}(z), (III.39)

where g_{k}(z) = ∑_{l≤k} g_{kl}z^{−l}. So, extracting the coefficient of z^{−l} in (III.39) we obtain exactly (III.31).

Finally, we state an important basic hypergeometric specialization of the matrix inversion in Theorem III.29.

Corollary III.40. Let

f_{nk} = ∏_{i=1}^{r} \frac{(ax_{i}q^{k_{i}−|k|}; q)_{n_{i}−k_{i}}}{(x_{i}q^{1+k_{i}+|k|}/a; q)_{n_{i}−k_{i}}} ∏_{i,j=1}^{r} \frac{(q^{k_{i}+k_{j}}x_{i}x_{j}; q)_{n_{i}−k_{i}}}{(q^{1+k_{i}−k_{j}}x_{i}/x_{j}; q)_{n_{i}−k_{i}}}  (III.41)

and

g_{kl} = (−1)^{|k|−|l|} q^{\binom{|k|−|l|}{2}} ∏_{1≤i≤j≤r} \frac{(1 − x_{i}x_{j}q^{l_{i}+l_{j}})}{(1 − x_{i}x_{j}q^{k_{i}+k_{j}})}

× ∏_{i=1}^{r} \frac{(ax_{i}q^{1+l_{i}−|k|}; q)_{k_{i}−l_{i}}}{(x_{i}q^{l_{i}+|k|}/a; q)_{k_{i}−l_{i}}} ∏_{i,j=1}^{r} \frac{(q^{1+l_{i}+k_{j}}x_{i}x_{j}; q)_{k_{i}−l_{i}}}{(q^{1+l_{i}−l_{j}}x_{i}/x_{j}; q)_{k_{i}−l_{i}}} . (III.42)

Then (f_{nk})_{n,k∈ℤ^{r}} and (g_{kl})_{k,l∈ℤ^{r}} are infinite lower-triangular r-dimensional matrices being inverses of each other.

Proof. In Theorem III.29 we set b ↦ 1/ax_{1} ⋯ x_{r}, and c_{i}(t_{i}) ↦ q^{−t_{i}}/x_{i} for i = 1, …, r. After some elementary manipulations which include Lemma B.1 we obtain the inverse pair (III.41) and (III.42).

Remark III.43. The inversion in Corollary III.40 is a C_{r} extension of Bressoud’s matrix inversion [16], to which it reduces for r = 1. But Corollary III.40 is also equivalent to the p = q case of Corollary III.25, with the roles of (f_{nk})_{n,k∈ℤ^{r}} and (g_{kl})_{k,l∈ℤ^{r}} being interchanged. Therefore, Corollary III.40 can also be viewed as a D_{r} basic hypergeometric matrix inversion. For an application of this inversion, see Theorem IV.22.

30 III. MULTIDIMENSIONAL MATRIX INVERSIONS

CHAPTER IV Multiple basic hypergeometric summation theorems

In this chapter, we use our multidimensional matrix inversions of chapter III for deriving new multiple basic hypergeometric summation theorems. First we explain the standard technique we apply, namely of the so-called inverse relations, (IV.1) and (IV.2). Then we recall the standard definitions in basic hypergeometric series theory and note the conventions we use in stating our multidimensional series. Finally, we give a number of new summation theorems.

Probably, the most important application of matrix inversion is the derivation of hypergeometric series identities. There is a standard technique for deriving new summation formulas from known ones by using inverse matrices (cf. [4], [29], [64]). If (f_{nk})_{n,k∈ℤ^{r}} and (g_{kl})_{k,l∈ℤ^{r}} are lower triangular matrices being inverses of each other, then of course the following is true:

∑_{0≤k≤n} f_{nk}a_{k} =b_{n}  (IV.1)

if and only if

∑_{0≤l≤k} g_{kl}b_{l} =a_{k}.  (IV.2)

Applications of our matrix inversions in Theorems III.7 and III.29 lead to many new identities for multidimensional (basic) hypergeometric series. As an illustration, we use special cases of our Theorem III.7 to derive A_{r} and D_{r} extensions of a terminating quadratic summation of Gessel and Stanton [29], D_{r} extensions of Jackson’s _{8}ϕ_{7} summation [40], and a D_{r} extension of a cubic summation of Gasper and Rahman [26].

We now recall the standard definitions in basic hypergeometric series theory (cf. [25]). Let q be a complex number such that |q| < 1. Define

(a; q)_{∞} := ∏_{j≥0}(1 − aq^{j}),

and,

(a; q)_{k} :=\frac{(a; q)_{∞}}{(aq^{k}; q)_{∞}}  (IV.3)

= ∏_{j=0}^{k−1}(1 − aq^{j}),  (IV.4)

where the equality (IV.4) holds when k is a non-negative integer. Classical (onedimensional) basic hypergeometric series (or q-hypergeometric series) with s numerator parameters a_{1}, …, a_{s} and t denominator parameters b_{1}, …, b_{t} are defined by

Definition IV.5 (_{s}ϕ_{t} basic hypergeometric series).

sϕ_{t}[\begin{matrix}a_{1}, …, a_{s} \\ b_{1}, …, b_{t} \end{matrix}; q, z] := ∑_{k=0}^{∞} \frac{(a_{1}; q)_{k} (a_{2}; q)_{k} ⋯ (a_{s}; q)_{k}}{(q; q)_{k} (b_{1}; q)_{k} ⋯ (b_{t}; q)_{k}} [(−1)^{k}q^{\binom{k}{2}}]^{1+t−s} z^{k}, (IV.6)

with \binom{k}{2} = k(k − 1)/2, where q ≠ 0 when s > t + 1. The parameters b_{1}, …, b_{t} are such that the denominator factors in the terms of the series (IV.6) are never zero.

Since (q^{−n}; q)_{k} = 0, if k = n + 1, n + 2, …, an _{s}ϕ_{t} series terminates if one of its numerator parameters is of the form q^{−n} with n = 0, 1, 2, …, and q ≠ 0. See [25] for the convergence criteria of these series when they do not terminate.

The theory of the (classical) basic hypergeometric series consists of several summation and transformation formulas involving _{s}ϕ_{t} series. Most of the fundamental summation and transformation formulas involve series where s = t + 1. The classical summation theorems for terminating _{3}ϕ_{2}, _{6}ϕ_{5}, and _{8}ϕ_{7} series require that the parameters satisfy the additional condition of being either balanced and/or very-well-poised. An _{s+1}ϕ_{s} basic hypergeometric series is called balanced if b_{1} ⋯ b_{s} = a_{1} ⋯ a_{s+1}q and z = q. An _{s+1}ϕ_{s} series is well-poised if a_{1}q = a_{2}b_{1} = ⋯ = a_{s+1}b_{s}. It is called very-well-poised if it is well-poised and if a_{2} = q\sqrt{a_{1}} and a_{3} = −q\sqrt{a_{1}}. Note that in a very-well-poised series, the factor

\frac{1 − a_{1}q^{2k}}{1 − a_{1}}  (IV.7)

appears in the summand of the series. The parameter a_{1} is usually referred to as the special parameter of the very-well-poised series.

One of the most powerful results in the theory and application of classical onedimensional basic hypergeometric series is Jackson’s [40] summation formula for a terminating _{8}ϕ_{7} series, which is both balanced and very-well-poised.

Theorem IV.8 (Jackson’s classical _{8}ϕ_{7} summation). Let a, b, c, and d be indeterminate, let n be a nonnegative integer and suppose that none of the denominators in (IV.9) vanish. Then

8ϕ_{7}[\begin{matrix}a, q\sqrt{a}, −q\sqrt{a}, b, c, d, a^{2}q^{n+1}/bcd, q^{−n} \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, bcdq^{−n}/a, aq^{n+1} \end{matrix}; q, q]

= \frac{(aq; q)_{n} (aq/bc; q)_{n} (aq/bd; q)_{n} (aq/cd; q)_{n}}{(aq/bcd; q)_{n} (aq/d; q)_{n} (aq/c; q)_{n} (aq/b; q)_{n}} . (IV.9)

Theorem IV.8 is equation (2.6.2) of [25], where we have chosen to do the replacement e → a^{2}q^{n+1}/bcd explicitly.

Remark IV.10. Limiting cases of Theorem IV.8 include various important summation theorems such as the nonterminating and terminating very-well-poised _{6}ϕ_{5} summations, the balanced q-Pfaff-Saalschütz _{3}ϕ_{2} summation, the nonterminating q-Gauß _{2}ϕ_{1} summation, and the terminating q-Chu-Vandermonde _{2}ϕ_{1} summation theorem. For more details, consult [25].

Before proceeding to multiple basic hypergeometric summation theorems, we explain what we mean by A_{r}, C_{r}, or D_{r} basic hypergeometric series. We consider multiple series of the form

∑_{{k_{i}≥0 \atop i=1,...,r}}S(k),  (IV.11)

where k = (k_{1}, …, k_{r}), which reduce to classical basic hypergeometric series when r = 1. We call such a multiple basic hypergeometric series balanced if it reduces to a balanced series when r = 1. Well-poised and very-well-poised series are defined similarly.

Further, such a multiple series is called a C_{r} basic hypergeometric series if the summand S(k) contains the factor

∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}+k_{j}}x_{i}x_{j}}{1 − x_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − x_{i}^{2}q^{2k_{i}}}{1 − x_{i}^{2}} ) .(IV.12)

Note that when r = 1, the first two factors disappear, and (IV.12) reduces to

\frac{1 − x_{1}^{2}q^{2k_{1}}}{1 − x_{1}^{2}} .  (IV.13)

The ratio (IV.13) is reminiscent of (IV.7). Indeed, in C_{r} series, x_{1}^{2} acts like the special parameter of a very-well-poised series. In our statements of C_{r} theorems, we often set x_{i} ↦ \sqrt{a}x_{i} for i = 1, …, r, and make similar changes to other parameters in S(k). This is done in order to follow the classical notation in [25] as closely as possible. A typical example of a C_{r} basic hypergeometric series is the left hand side of (B.7).

D_{r} multiple basic series are closely related to C_{r} series. Instead of (IV.12), S(k) only has the following factors:

∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}+k_{j}}x_{i}x_{j}}{1 − x_{i}x_{j}} ) .  (IV.14)

A typical example is the left hand side of (IV.23), see Remark IV.25. Finally, A_{r} basic hypergeometric series only have

∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )  (IV.15)

as a factor of S(k). A typical example is the left hand side of (IV.17). The other types of products occurring in the three types of series are similar. A reason for naming these series as A_{r}, C_{r}, or D_{r} series is that (IV.15), (IV.12), and (IV.14) are closely associated with the product side of the Weyl denominator formulas for the respective root systems, see Lemma VII.45 (cf. [72], [22], and [12]).

We terminate our multiple series in two ways: If the summand contains the factor

∏_{i,j=1}^{r}(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}

in the numerator, then it vanishes if k_{i} > n_{i} for i = 1, …, r. In this case we have the natural bounds 0 ≤ k_{i} ≤ n_{i}, for i = 1, …, r, and we say that the series is summed over an r-rectangle. On the other hand, if the summand contains the factor

(q^{−N}; q)_{k_{1}+···+k_{r}}

in the numerator, then it vanishes if k_{1} + ⋯ + k_{r} > N. In this case we have the natural bounds 0 ≤ k_{1} + ⋯ + k_{r} ≤ N for all k_{i} ≥ 0, and we say that the series is summed over an r-tetrahedron. Many identities involving series summed over an r-rectangle are equivalent to a corresponding identity, where the series involved are summed over an r-tetrahedron, see e.g. Theorem IV.16 and Theorem IV.20, or Theorem IV.22 and Theorem IV.26.

Finally, we mention some notation used to simplify our displays. We employ the notation |k| for (k_{1} + ⋯ + k_{r}), where k = (k_{1}, …, k_{r}). This notation is also applied to other vectors. We also find it useful to denote by m + k the vector (m_{1} + k_{1}, …, m_{r} + k_{r}) formed by component-wise addition of the two vectors.

Now we can present our multiple basic hypergeometric summation theorems.

Theorem IV.16 (An A_{r} quadratic sum). Let x_{1}, …, x_{r}, a, b, and d be indeterminate, let n_{1}, …, n_{r} be nonnegative integers, let r ≥ 1, and suppose that none of the denominators in (IV.17) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{2k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{2k_{i}−2k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(q^{−2n_{j}}x_{i}/x_{j}; q^{2})_{k_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{k_{i}}}

× ∏_{i=1}^{r} \frac{(dx_{i}; q^{2})_{k_{i}} (a^{2}x_{i}q^{1+2|n|}/d; q^{2})_{k_{i}}}{(ax_{i}q^{2}/b; q^{2})_{k_{i}} (abx_{i}q; q^{2})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q^{1+2n_{i}}; q)_{|k|}}

× \frac{(b; q)_{|k|} (q/b; q)_{|k|}}{(aq/d; q)_{|k|} (dq^{−2|n|}/a; q)_{|k|}} q^{−|k|+2 ∑_{i=1}^{r} i k_{i}} )

= \frac{(aq^{2}/bd; q^{2})_{|n|} (abq/d; q^{2})_{|n|}}{(aq/d; q)_{2|n|}} ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{2n_{i}}}{(ax_{i}q^{2}/b; q^{2})_{n_{i}} (abx_{i}q; q^{2})_{n_{i}}} . (IV.17)

Remark IV.18. This quadratic summation formula is an A_{r} extension of

∑_{k=0}^{n} \frac{1 − aq^{3k}}{1 − a} \frac{(a; q)_{k}(b; q)_{k}(q/b; q)_{k} (d; q^{2})_{k}(a^{2}q^{1+2n}/d; q^{2})_{k}(q^{−2n}; q^{2})_{k}}{(q^{2}; q^{2})_{k}(aq^{2}/b; q^{2})_{k}(abq; q^{2})_{k} (aq/d; q)_{k}(dq^{−2n}/a; q)_{k}(aq^{2n+1}; q)_{k}} q^{k}

= \frac{(aq; q)_{2n}}{(aq/d; q)_{2n}} \frac{(abq/d; q^{2})_{n}(aq^{2}/bd; q^{2})_{n}}{(aq^{2}/b; q^{2})_{n}(abq; q^{2})_{n}} , (IV.19)

due to Gessel and Stanton [29, eq. (1.4), q → q^{2}], to which it reduces for r = 1. Many identities like (IV.19), involving bases of different powers of q, are known. Hypergeometric series with several bases were extensively studied by Gasper and Rahman [24], [63], [26], [25, sec. 3.8].

Proof of Theorem IV.16. If we substitute q ↦ q^{2}, p ↦ q, and d ↦ a^{2}/d in Corollary III.21 we see that the following pair of matrices are inverses of each other:

f_{nk} = ∏_{i=1}^{r} (\frac{1 − q^{1+2k_{i}+2|k|}a^{2}x_{i}/d}{1 − qa^{2}x_{i}/d}) ∏_{1≤i<j≤r} (\frac{1 − q^{2k_{i}−2k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−2n_{j}}x_{i}/x_{j}; q^{2})_{k_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}q^{|n|}; q)_{2k_{i}}}{(a^{2}x_{i}q^{3+2n_{i}}/d; q^{2})_{|k|}} · \frac{q^{2 ∑_{i=1}^{r} i k_{i}}}{(aq^{2−|n|}/d; q)_{2|k|}}

and

g_{kl} = ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{2l_{i}+|l|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{2l_{i}−2l_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(q^{−2k_{j}}x_{i}/x_{j}; q^{2})_{l_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{l_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|l|}}{(ax_{i}q^{1+2k_{i}}; q)_{|l|}} ∏_{i=1}^{r} \frac{(a^{2}x_{i}q^{1+2|k|}/d; q^{2})_{l_{i}}}{(a^{2}x_{i}q^{3}/d; q^{2})_{l_{i}}} ∏_{i=1}^{r} \frac{(a^{2}x_{i}q/d; q^{2})_{|k|}}{(ax_{i}q; q)_{2k_{i}}}

× \frac{(1 − q^{1+|l|}a/d)}{(1 − qa/d)} \frac{(d/aq; q)_{|l|} (qa/d; q)_{2|k|}}{(q^{2−2|k|}a/d; q)_{|l|}} q^{−|l|+2 ∑_{i=1}^{r} i l_{i}}.

Now (IV.1) holds for

a_{k} = (baq/d; q^{2})_{|k|} (aq^{2}/bd; q^{2})_{|k|} ∏_{i=1}^{r} \frac{(a^{2}x_{i}q/d; q^{2})_{|k|}}{(ax_{i}q^{2}/b; q^{2})_{k_{i}} (abx_{i}q; q)_{k_{i}}}

and

b_{n} = \frac{(q^{2−|n|}/b; q^{2})_{|n|} (bq^{1−|n|}; q^{2})_{|n|}}{(aq^{3−|n|}/d; q^{2})_{|n|} (dq^{−|n|}/a; q^{2})_{|n|}} ∏_{i=1}^{r} \frac{(a^{2}x_{i}q^{3}/d; q^{2})_{n_{i}} (dx_{i}/a; q^{2})_{n_{i}}}{(ax_{i}q^{2}/b; q^{2})_{n_{i}} (abx_{i}q; q^{2})_{n_{i}}}

by means of an A_{r} extension of Jackson’s _{8}ϕ_{7}-sum, Theorem B.3. This implies the inverse relation (IV.2) which is easily transformed into (IV.17).

It is not hard to see from a polynomial identity argument that Theorem IV.16 implies the following summation theorem.

Theorem IV.20 (An A_{r} quadratic sum). Let x_{1}, …, x_{r}, c_{1}, …, c_{r}, a, and d be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (IV.21) vanish. Then

∑_{k_{1},...,k_{r}≥0} ; 0≤|k|≤N( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{2k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{2k_{i}−2k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(c_{j}x_{i}/x_{j}; q^{2})_{k_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{k_{i}}}

× ∏_{i=1}^{r} \frac{(dx_{i}; q^{2})_{k_{i}} (a^{2}x_{i}q/d ∏_{j=1}^{r} c_{j}; q^{2})_{k_{i}}}{(ax_{i}q^{2+N}; q^{2})_{k_{i}} (ax_{i}q^{1−N}; q^{2})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q/c_{i}; q)_{|k|}}

× \frac{(q^{−N}; q)_{|k|} (q^{1+N}; q)_{|k|}}{(aq/d; q)_{|k|} (d ∏_{j=1}^{r} c_{j}/a; q)_{|k|}} q^{−|k|+2 ∑_{i=1}^{r} i k_{i}} )

= \begin{cases}\frac{(dq/a; q^{2})_{M}(aq^{2}/d ∏_{j=1}^{r} c_{j}; q^{2})_{M}}{(aq^{2}/d; q^{2})_{M}(dq ∏_{j=1}^{r} c_{j}/a; q^{2})_{M}} ∏_{i=1}^{r} \frac{(ax_{i}q^{2}; q^{2})_{M}(c_{i}q/ax_{i}; q^{2})_{M}}{(q/ax_{i}; q^{2})_{M}(ax_{i}q^{2}/c_{i}; q^{2})_{M}} & (N = 2M), \\ \frac{(d/a; q^{2})_{M}(aq/d ∏_{j=1}^{r} c_{j}; q^{2})_{M}}{(aq/d; q^{2})_{M}(d ∏_{j=1}^{r} c_{j}/a; q^{2})_{M}} ∏_{i=1}^{r} \frac{(ax_{i}q; q^{2})_{M}(c_{i}/ax_{i}; q^{2})_{M}}{(1/ax_{i}; q^{2})_{M}(ax_{i}q/c_{i}; q^{2})_{M}} & (N = 2M − 1). \end{cases}

(IV.21)

Proof. First we write the right sides of (IV.21) as quotients of infinite products using (IV.3). Then by the b = q^{−N} case of Theorem IV.16 it follows that the identity (IV.21) holds for c_{j} = q^{−2n_{j}}, j = 1, …, r. By clearing out denominators in (IV.21), we get a polynomial equation in c_{1}, which is true for q^{−2n_{1}}, n_{1} = 0, 1, . . .. Thus we obtain an identity in c_{1}. By carrying out this process for c_{2}, c_{3}, …, c_{r} also, we obtain Theorem IV.20.

In chapter III, we obtained by specialization an interesting bibasic hypergeometric matrix inversion, Corollary III.25. We use this inversion to derive D_{r} basic hypergeometric summation formulas.

Theorem IV.22 (A D_{r} Jackson’s sum). Let x_{1}, …, x_{r}, a, b, and c be indeterminate, let n_{1}, …, n_{r} be nonnegative integers, let r ≥ 1, and suppose that none of the denominators in (IV.23) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}; q)_{k_{i}+k_{j}}^{−1}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (x_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/x_{i}; q)_{|k|−k_{i}}}{(aq^{1+n_{i}}x_{i}; q)_{|k|} (aq^{1−n_{i}}/x_{i}; q)_{|k|}}

× \frac{(b; q)_{|k|} (c; q)_{|k|} (a^{2}q/bc; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (bcx_{i}/a; q)_{k_{i}}]} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/bc; q)_{n_{i}} (bx_{i}/a; q)_{n_{i}} (cx_{i}/a; q)_{n_{i}}}{(x_{i}/a; q)_{n_{i}} (bcx_{i}/a; q)_{n_{i}} (ax_{i}q/b; q)_{n_{i}} (ax_{i}q/c; q)_{n_{i}}} . (IV.23)

Remark IV.24. For r = 1 Theorem IV.22 reduces to Jackson’s balanced verywell-poised _{8}ϕ_{7} summation formula [40], [25, (II.22)], which is displayed in (IV.9). Theorem IV.22 is stated as Theorem B.12 in Appendix B again, where the notation is intended to follow Gasper-Rahman as closely as possible. In chapter V we use Theorem B.12 to derive multiple extensions of Bailey’s [7] balanced very-well-poised _{10}ϕ_{9} transformation.

Remark IV.25. We view the left hand side of (IV.23) as a D_{r} series since the summand contains the factor

∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}; q)_{k_{i}+k_{j}}^{−1}

= ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}+k_{j}}x_{i}x_{j}}{1 − x_{i}x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}q; q)_{k_{i}+k_{j}}^{−1}.

As the series contains the factor (IV.14), at least in disguise, according to our conventions it belongs to type D_{r}. Also note that A_{r} series do not contain the factor ∏_{1≤i<j≤r}(x_{i}x_{j}; q)_{k_{i}+k_{j}}^{−1} in the summand.

Proof of Theorem IV.22. By Corollary III.40 (or by setting p = q in Corollary III.25, i.e. we consider a C_{r}/D_{r} extension of Bressoud’s matrix inverse [16]) we see that the following pair of matrices are inverses of each other:

f_{nk} = ∏_{1≤i<j≤r} \frac{(1 − q^{k_{i}−k_{j}}x_{i}/x_{j})(1 − x_{i}x_{j}q^{k_{i}+k_{j}})}{(1 − x_{i}/x_{j})(1 − x_{i}x_{j})} ∏_{i=1}^{r} \frac{(1 − x_{i}^{2}q^{2k_{i}})}{(1 − x_{i}^{2})}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (x_{i}x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (x_{i}x_{j}q^{1+n_{j}}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}q^{|n|}; q)_{k_{i}}}{(x_{i}q^{1−|n|}/a; q)_{k_{i}}} · q^{∑_{i=1}^{r} i k_{i}}

and

g_{kl} = ∏_{1≤i<j≤r} \frac{(1 − q^{l_{i}−l_{j}}x_{i}/x_{j})(1 − x_{i}x_{j}q^{l_{i}+l_{j}})}{(1 − x_{i}/x_{j})(1 − x_{i}x_{j})}

× ∏_{i,j=1}^{r} \frac{(q^{−k_{j}}x_{i}/x_{j}; q)_{l_{i}} (x_{i}x_{j}q^{k_{j}}; q)_{l_{i}}}{(qx_{i}/x_{j}; q)_{l_{i}} (x_{i}x_{j}q; q)_{l_{i}}} ∏_{i=1}^{r} \frac{(1 − aq^{|l|+l_{i}}x_{i})(1 − aq^{|l|−l_{i}}/x_{i})}{(1 − ax_{i})(1 − a/x_{i})}

× ∏_{i=1}^{r} \frac{(x_{i}/a; q)_{k_{i}}}{(ax_{i}q; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|l|} (a/x_{i}; q)_{|l|}}{(aq^{1+k_{i}}x_{i}; q)_{|l|} (aq^{1−k_{i}}/x_{i}; q)_{|l|}} · q^{∑_{i=1}^{r} i l_{i}}.

Now (IV.1) holds for

a_{k} = ∏_{i=1}^{r} \frac{(bx_{i}/a; q)_{k_{i}} (cx_{i}/a; q)_{k_{i}} (ax_{i}q/bc; q)_{k_{i}}}{(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (bcx_{i}/a; q)_{k_{i}}}

and

b_{n} = ∏_{i=1}^{r} \frac{(x_{i}^{2}q; q)_{n_{i}}}{(bcx_{i}/a; q)_{n_{i}} (x_{i}q^{1−|n|}/a; q)_{n_{i}}} ∏_{1≤i<j≤r} \frac{(x_{i}x_{j}q; q)_{n_{i}}}{(x_{i}x_{j}q^{1+n_{j}}; q)_{n_{i}}}

× \frac{(bcq^{−|n|}/a^{2}; q)_{|n|} (q^{1−|n|}/b; q)_{|n|} (c; q)_{|n|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{n_{i}} (cq^{−n_{i}}/ax_{i}; q)_{n_{i}}]}

by Denis and Gustafson, Milne and Lilly’s C_{r} _{8}ϕ_{7} summation B.6. This implies the inverse relation (IV.2) which is easily transformed into (IV.23).

By using a polynomial argument we get

Theorem IV.26 (A D_{r} Jackson’s sum). Let x_{1}, …, x_{r}, c_{1}, …, c_{r}, a, and b be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (IV.27) vanish. Then

∑_{k_{1},...,k_{r}≥0} ; 0≤|k|≤N( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}; q)_{k_{i}+k_{j}}^{−1}

× ∏_{i,j=1}^{r} \frac{(c_{j}x_{i}/x_{j}; q)_{k_{i}} (x_{i}x_{j}/c_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/x_{i}; q)_{|k|−k_{i}}}{(ax_{i}q/c_{i}; q)_{|k|} (ac_{i}q/x_{i}; q)_{|k|}}

× \frac{(b; q)_{|k|} (a^{2}q^{1+N}/b; q)_{|k|} (q^{−N}; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{k_{i}} (bx_{i}q^{−N}/a; q)_{k_{i}} (ax_{i}q^{1+N}; q)_{k_{i}}]} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{N} (aq/x_{i}; q)_{N} (ax_{i}q/bc_{i}; q)_{N} (ac_{i}q/bx_{i}; q)_{N}}{(aq/bx_{i}; q)_{N} (ax_{i}q/b; q)_{N} (ac_{i}q/x_{i}; q)_{N} (ax_{i}q/c_{i}; q)_{N}} . (IV.27)

Proof. First we write the right side of (IV.27) as quotient of infinite products using (IV.3). Then by the c = q^{−N} case of Theorem IV.22 it follows that the identity (IV.27) holds for c_{j} = q^{−n_{j}}, j = 1, …, r. By clearing out denominators in (IV.27), we get a polynomial equation in c_{1}, which is true for q^{−n_{1}}, n_{1} = 0, 1, . . .. Thus we obtain an identity in c_{1}. By carrying out this process for c_{2}, c_{3}, …, c_{r} also, we obtain Theorem IV.26.

Limiting cases of Theorem IV.22 or Theorem IV.26 include various D_{r} summations. By reversing the multisum in Theorem IV.22 we obtain another D_{r} Jackson’s sum, Theorem B.9, which was independently derived by G. Bhatnagar [12] using a different method. D_{r} extensions of many of the classical basic hypergeometric summation theorems are given in [12]. Further consequences of the new D_{r} 8ϕ_{7} summations, such as C_{r} and D_{r} extensions of Bailey’s very-well-poised _{10}ϕ_{9} transformation formula [7], [25, (III.28)] are given in chapter V.

Remark IV.28. We note that (IV.23) and (IV.27) could be written (with a = 1/x_{r+1} and k_{r+1} := −|k|) more compactly as

∑_{k_{1}+···+k_{r+1}=0} ( ∏_{1≤i<j≤r+1} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r+1}(x_{i}x_{j}; q)_{k_{i}+k_{j}}^{−1}

× ∏_{i=1}^{r+1} ∏_{j=1}^{r} \frac{(c_{j}x_{i}/x_{j}; q)_{k_{i}} (x_{i}x_{j}/c_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r+1} [(x_{i}q/bx_{r+1}; q)_{k_{i}} (x_{i}q/cx_{r+1}; q)_{k_{i}} (bcx_{i}x_{r+1}; q)_{k_{i}}]^{−1}

× ∏_{i=1}^{r+1} x_{i}^{k_{i}} · q^{∑{r+1 \atop i=1}[\binom{k_{i}}{2}+i k_{i}]})

= ∏_{i=1}^{r} \frac{(x_{i}q/x_{r+1}; q)_{∞}(q/x_{i}x_{r+1}; q)_{∞}(x_{i}q/bc_{i}x_{r+1}; q)_{∞}(c_{i}q/bx_{i}x_{r+1}; q)_{∞}}{(q/bx_{i}x_{r+1}; q)_{∞}(x_{i}q/bx_{r+1}; q)_{∞}(c_{i}q/x_{i}x_{r+1}; q)_{∞}(x_{i}q/c_{i}x_{r+1}; q)_{∞}}

× ∏_{i=1}^{r} \frac{(q/bcx_{i}x_{r+1}; q)_{∞}(x_{i}q/bcx_{r+1}; q)_{∞}(c_{i}q/cx_{i}x_{r+1}; q)_{∞}(x_{i}q/cc_{i}x_{r+1}; q)_{∞}}{(x_{i}q/cx_{r+1}; q)_{∞}(q/cx_{i}x_{r+1}; q)_{∞}(x_{i}q/bcc_{i}x_{r+1}; q)_{∞}(c_{i}q/bcx_{i}x_{r+1}; q)_{∞}} ,

(IV.29)

provided the series terminates. However, we feel that the forms (IV.23) and (IV.27) are preferable since the dependence of summation indices in (IV.29) (which just hides what is really going on in the sum) is removed.

The remainder of this chapter is devoted to D_{r} quadratic and cubic summation theorems.

Theorem IV.30 (A D_{r} quadratic sum). Let x_{1}, …, x_{r}, a, and b be indeterminate, let n_{1}, …, n_{r} be nonnegative integers, let r ≥ 1, and suppose that none of the denominators in (IV.31) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{2k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{2k_{i}−2k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}; q^{2})_{k_{i}+k_{j}}^{−1}

× ∏_{i,j=1}^{r} \frac{(q^{−2n_{j}}x_{i}/x_{j}; q^{2})_{k_{i}} (x_{i}x_{j}q^{2n_{j}}; q^{2})_{k_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/x_{i}; q)_{|k|−2k_{i}}}{(ax_{i}q^{1+2n_{i}}; q)_{|k|} (aq^{1−2n_{i}}/x_{i}; q)_{|k|}}

× \frac{(a^{2}q; q^{2})_{|k|} (b; q)_{|k|} (q/b; q)_{|k|}}{∏_{i=1}^{r} [(abx_{i}q; q^{2})_{k_{i}} (ax_{i}q^{2}/b; q^{2})_{k_{i}}]} (−a)^{|k|} ∏_{i=1}^{r} x_{i}^{−k_{i}} · q^{2e_{2}(k)−\binom{|k|+1}{2}+2 ∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{2n_{i}} (x_{i}q/ab; q^{2})_{n_{i}} (bx_{i}/a; q^{2})_{n_{i}}}{(x_{i}/a; q)_{2n_{i}} (abx_{i}q; q^{2})_{n_{i}} (ax_{i}q^{2}/b; q^{2})_{n_{i}}} , (IV.31)

where e_{2}(k) is the second elementary symmetric function of {k_{1}, …, k_{r}}.

Remark IV.32. This quadratic summation formula is a D_{r} extension of Gessel and Stanton’s summation [29, eq. (1.4)], displayed in (IV.19), to which it reduces for r = 1.

Proof of Theorem IV.30. Doing the replacements q → q^{2}, p → q in Corollary III.25 we see that the following pair of matrices are inverses of each other:

f_{nk} = ∏_{1≤i<j≤r} \frac{(1 − q^{2k_{i}−2k_{j}}x_{i}/x_{j})(1 − x_{i}x_{j}q^{2k_{i}+2k_{j}})}{(1 − x_{i}/x_{j})(1 − x_{i}x_{j})} ∏_{i=1}^{r} \frac{(1 − x_{i}^{2}q^{4k_{i}})}{(1 − x_{i}^{2})}

× ∏_{i,j=1}^{r} \frac{(q^{−2n_{j}}x_{i}/x_{j}; q^{2})_{k_{i}} (x_{i}x_{j}; q^{2})_{k_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{k_{i}} (x_{i}x_{j}q^{2+2n_{j}}; q^{2})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}q^{|n|}; q)_{2k_{i}}}{(x_{i}q^{1−|n|}/a; q)_{2k_{i}}} · q^{2 ∑_{i=1}^{r} i k_{i}}

and

g_{kl} = ∏_{1≤i<j≤r} \frac{(1 − q^{2l_{i}−2l_{j}}x_{i}/x_{j})(1 − x_{i}x_{j}q^{2l_{i}+2l_{j}})}{(1 − x_{i}/x_{j})(1 − x_{i}x_{j})}

× ∏_{i,j=1}^{r} \frac{(q^{−2k_{j}}x_{i}/x_{j}; q^{2})_{l_{i}} (x_{i}x_{j}q^{2k_{j}}; q^{2})_{l_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{l_{i}} (x_{i}x_{j}q^{2}; q^{2})_{l_{i}}} ∏_{i=1}^{r} \frac{(1 − aq^{|l|+2l_{i}}x_{i})(1 − aq^{|l|−2l_{i}}/x_{i})}{(1 − ax_{i})(1 − a/x_{i})}

× ∏_{i=1}^{r} \frac{(x_{i}/a; q)_{2k_{i}}}{(ax_{i}q; q)_{2k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|l|} (a/x_{i}; q)_{|l|}}{(aq^{1+2k_{i}}x_{i}; q)_{|l|} (aq^{1−2k_{i}}/x_{i}; q)_{|l|}} · q^{2 ∑_{i=1}^{r} i l_{i}}.

Now (IV.1) holds for

a_{k} = ∏_{i=1}^{r} \frac{(x_{i}q/ab; q^{2})_{k_{i}} (bx_{i}/a; q^{2})_{k_{i}}}{(abx_{i}q; q^{2})_{k_{i}} (ax_{i}q^{2}/b; q^{2})_{k_{i}}}

and

b_{n} = ∏_{i=1}^{r} \frac{(x_{i}^{2}q^{2}; q^{2})_{n_{i}}}{(abx_{i}q; q^{2})_{n_{i}} (x_{i}q^{2−|n|}/a; q^{2})_{n_{i}}} ∏_{1≤i<j≤r} \frac{(x_{i}x_{j}q^{2}; q^{2})_{n_{i}}}{(x_{i}x_{j}q^{2+2n_{j}}; q^{2})_{n_{i}}}

× \frac{(bq^{1−|n|}; q^{2})_{|n|} (q^{2−|n|}/b; q^{2})_{|n|} (a^{2}q; q^{2})_{|n|}}{∏_{i=1}^{r} [(ax_{i}q^{2}/b; q^{2})_{n_{i}} (aq^{1+|n|−2n_{i}}/x_{i}; q^{2})_{n_{i}}]}

by Denis and Gustafson, Milne and Lilly’s C_{r} _{8}ϕ_{7} summation B.6. This implies the inverse relation (IV.2) which is easily transformed into (IV.31).

Remark IV.33. By reversing the multisum in (IV.31) we may obtain another, differently looking, extension of Gessel and Stanton’s quadratic summation formula (IV.19).

Using the same technique as in the proofs of Theorems IV.20 and IV.26 we obtain from Theorem IV.30 the next two quadratic summation theorems (with minor substitutions of variables in Theorem IV.36).

Theorem IV.34 ( A D_{r} quadratic sum). Let x_{1}, …, x_{r}, c_{1}, …, c_{r}, and a be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (IV.35) vanish. Then

∑_{k_{1},...,k_{r}≥0} ; 0≤|k|≤N( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{2k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{2k_{i}−2k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}; q^{2})_{k_{i}+k_{j}}^{−1}

× ∏_{i,j=1}^{r} \frac{(c_{j}x_{i}/x_{j}; q^{2})_{k_{i}} (x_{i}x_{j}/c_{j}; q^{2})_{k_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/x_{i}; q)_{|k|−2k_{i}}}{(ax_{i}q/c_{i}; q)_{|k|} (ac_{i}q/x_{i}; q)_{|k|}}

× \frac{(a^{2}q; q^{2})_{|k|} (q^{1+N}; q)_{|k|} (q^{−N}; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q^{2+N}; q^{2})_{k_{i}} (ax_{i}q^{1−N}; q^{2})_{k_{i}}]} (−a)^{|k|} ∏_{i=1}^{r} x_{i}^{−k_{i}} · q^{2e_{2}(k)−\binom{|k|+1}{2}+2 ∑_{i=1}^{r} i k_{i}} )

= \begin{cases}∏_{i=1}^{r} \frac{(ax_{i}q^{2}; q^{2})_{M} (aq^{2}/x_{i}; q^{2})_{M} (c_{i}q/ax_{i}; q^{2})_{M} (x_{i}q/ac_{i}; q^{2})_{M}}{(q/ax_{i}; q^{2})_{M} (x_{i}q/a; q^{2})_{M} (ax_{i}q^{2}/c_{i}; q^{2})_{M} (ac_{i}q^{2}/x_{i}; q^{2})_{M}} & (N = 2M), \\ ∏_{i=1}^{r} \frac{(ax_{i}q; q^{2})_{M} (aq/x_{i}; q^{2})_{M} (c_{i}/ax_{i}; q^{2})_{M} (x_{i}/ac_{i}; q^{2})_{M}}{(1/ax_{i}; q^{2})_{M} (x_{i}/a; q^{2})_{M} (ax_{i}q/c_{i}; q^{2})_{M} (ac_{i}q/x_{i}; q^{2})_{M}} & (N = 2M − 1), \end{cases}

(IV.35)

where e_{2}(k) is the second elementary symmetric function of {k_{1}, …, k_{r}}.

Theorem IV.36 (A D_{r} quadratic sum). Let x_{1}, …, x_{r}, c_{1}, …, c_{r}, and b be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (IV.37) vanish. Then

∑_{k_{1},...,k_{r}≥0} ; 0≤|k|≤N( ∏_{i=1}^{r}(\frac{1 − x_{i}q^{2k_{i}+|k|}}{1 − x_{i}} ) ∏_{1≤i<j≤r}(\frac{1 − q^{2k_{i}−2k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}q^{1+2N}; q^{2})_{k_{i}+k_{j}}^{−1}

× ∏_{i,j=1}^{r} \frac{(c_{j}x_{i}/x_{j}; q^{2})_{k_{i}} (x_{i}x_{j}q^{1+2N}/c_{j}; q^{2})_{k_{i}}}{(q^{2}x_{i}/x_{j}; q^{2})_{k_{i}}} ∏_{i=1}^{r} \frac{(x_{i}; q)_{|k|} (q^{−2N}/x_{i}; q)_{|k|−2k_{i}}}{(x_{i}q/c_{i}; q)_{|k|} (c_{i}q^{−2N}/x_{i}; q)_{|k|}}

× \frac{(b; q)_{|k|} (q/b; q)_{|k|} (q^{−2N}; q^{2})_{|k|}}{∏_{i=1}^{r} [(bx_{i}q; q^{2})_{k_{i}} (x_{i}q^{2}/b; q^{2})_{k_{i}}]} ∏_{i=1}^{r}(−x_{i})^{−k_{i}} · q^{2e_{2}(k)−\binom{|k|}{2}−2N|k|+2 ∑_{i=1}^{r}(i−1)k_{i}} )

= ∏_{i=1}^{r} \frac{(x_{i}q; q)_{2N} (bx_{i}q/c_{i}; q^{2})_{N} (x_{i}q^{2}/bc_{i}; q^{2})_{N}}{(x_{i}q/c_{i}; q)_{2N} (x_{i}q^{2}/b; q^{2})_{N} (bx_{i}q; q^{2})_{N}} , (IV.37)

where e_{2}(k) is the second elementary symmetric function of {k_{1}, …, k_{r}}.

Finally, we derive some cubic summations.

Theorem IV.38 (A D_{r} cubic sum). Let x_{1}, …, x_{r}, and a be indeterminate, let n_{1}, …, n_{r} be nonnegative integers, let r ≥ 1, and suppose that none of the denominators in (IV.39) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{3k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{3k_{i}−3k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}; q^{3})_{k_{i}+k_{j}}^{−1}

IV. MULTIPLE BASIC HYPERGEOMETRIC SUMMATION THEOREMS 43

× ∏_{i,j=1}^{r} \frac{(q^{−3n_{j}}x_{i}/x_{j}; q^{3})_{k_{i}} (x_{i}x_{j}q^{3n_{j}}; q^{3})_{k_{i}}}{(q^{3}x_{i}/x_{j}; q^{3})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/x_{i}; q)_{|k|−3k_{i}}}{(ax_{i}q^{1+3n_{i}}; q)_{|k|} (aq^{1−3n_{i}}/x_{i}; q)_{|k|}}

× \frac{(1/a^{2}; q)_{|k|} (a^{2}q; q)_{2|k|}}{∏_{i=1}^{r}(a^{3}x_{i}q^{3}; q^{3})_{k_{i}}} a^{2|k|} ∏_{i=1}^{r} x_{i}^{−2k_{i}} · q^{6e_{2}(k)−\binom{2|k|+1}{2}+3 ∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{3n_{i}} (x_{i}/a^{3}; q^{3})_{n_{i}}}{(x_{i}/a; q)_{3n_{i}} (a^{3}x_{i}q^{3}; q^{3})_{n_{i}}} , (IV.39)

where e_{2}(k) is the second elementary symmetric function of {k_{1}, …, k_{r}}. Remark IV.40. This cubic summation formula is a D_{r} extension of

∑_{k=0}^{m} \frac{1 − aq^{4k}}{1 − a} \frac{(a; q)_{k} (b; q)_{k} (q/b; q)_{2k} (a^{2}bq^{3n}; q^{3})_{k} (q^{−3n}; q^{3})_{k}}{(q^{3}; q^{3})_{k} (aq^{3}/b; q^{3})_{k} (ab; q)_{2k} (q^{1−3n}/ab; q)_{k} (aq^{3n+1}; q)_{k}} q^{k}

= \frac{(aq; q)_{3n}}{(ab; q)_{3n}} \frac{(ab^{2}; q^{3})_{n}}{(aq^{3}/b; q^{3})_{n}} , (IV.41)

due to Gasper and Rahman [26, eq. (4.1), c → 1], to which it reduces for r = 1.

Proof of Theorem IV.38. Doing the replacements q → q^{3}, p → q in Corollary III.25 we see that the following pair of matrices are inverses of each other:

f_{nk} = ∏_{1≤i<j≤r} \frac{(1 − q^{3k_{i}−3k_{j}}x_{i}/x_{j})(1 − x_{i}x_{j}q^{3k_{i}+3k_{j}})}{(1 − x_{i}/x_{j})(1 − x_{i}x_{j})} ∏_{i=1}^{r} \frac{(1 − x_{i}^{2}q^{6k_{i}})}{(1 − x_{i}^{2})}

× ∏_{i,j=1}^{r} \frac{(q^{−3n_{j}}x_{i}/x_{j}; q^{3})_{k_{i}} (x_{i}x_{j}; q^{3})_{k_{i}}}{(q^{3}x_{i}/x_{j}; q^{3})_{k_{i}} (x_{i}x_{j}q^{3+3n_{j}}; q^{3})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}q^{|n|}; q)_{3k_{i}}}{(x_{i}q^{1−|n|}/a; q)_{3k_{i}}} · q^{3 ∑_{i=1}^{r} i k_{i}}

and

g_{kl} = ∏_{1≤i<j≤r} \frac{(1 − q^{3l_{i}−3l_{j}}x_{i}/x_{j})(1 − x_{i}x_{j}q^{3l_{i}+3l_{j}})}{(1 − x_{i}/x_{j})(1 − x_{i}x_{j})}

× ∏_{i,j=1}^{r} \frac{(q^{−3k_{j}}x_{i}/x_{j}; q^{3})_{l_{i}} (x_{i}x_{j}q^{3k_{j}}; q^{3})_{l_{i}}}{(q^{3}x_{i}/x_{j}; q^{3})_{l_{i}} (x_{i}x_{j}q^{3}; q^{3})_{l_{i}}} ∏_{i=1}^{r} \frac{(1 − aq^{|l|+3l_{i}}x_{i})(1 − aq^{|l|−3l_{i}}/x_{i})}{(1 − ax_{i})(1 − a/x_{i})}

× ∏_{i=1}^{r} \frac{(x_{i}/a; q)_{3k_{i}}}{(ax_{i}q; q)_{3k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|l|} (a/x_{i}; q)_{|l|}}{(aq^{1+3k_{i}}x_{i}; q)_{|l|} (aq^{1−3k_{i}}/x_{i}; q)_{|l|}} · q^{3 ∑_{i=1}^{r} i l_{i}}.

Now (IV.1) holds for

a_{k} = ∏_{i=1}^{r} \frac{(x_{i}/a^{3}; q^{3})_{k_{i}}}{(a^{3}x_{i}q^{3}; q^{3})_{k_{i}}}

and

b_{n} = ∏_{i=1}^{r} \frac{(x_{i}^{2}q^{3}; q^{3})_{n_{i}}}{(x_{i}q^{1−|n|}/a; q^{3})_{n_{i}} (x_{i}q^{3−|n|}/a; q^{3})_{n_{i}}} ∏_{1≤i<j≤r} \frac{(x_{i}x_{j}q^{3}; q^{3})_{n_{i}}}{(x_{i}x_{j}q^{3+3n_{j}}; q^{3})_{n_{i}}}

× \frac{(q^{1−2|n|}/a^{2}; q^{3})_{|n|} (a^{2}q^{3−|n|}; q^{3})_{|n|} (a^{2}q^{1−|n|}; q^{3})_{|n|}}{∏_{i=1}^{r} [(a^{3}x_{i}q^{3}; q^{3})_{n_{i}} (aq^{1+|n|−3n_{i}}/x_{i}; q^{3})_{n_{i}}]}

by Denis and Gustafson, Milne and Lilly’s C_{r} _{8}ϕ_{7} summation B.6. This implies the inverse relation (IV.2) which is easily transformed into (IV.39).

Remark IV.42. By reversing the multisum in (IV.39) we may obtain another, differently looking, D_{r} extension of Gasper and Rahman’s cubic summation formula (IV.41).

Using the same polynomial argument as above, we derive some more D_{r} cubic summation theorems from Theorem IV.38. For sake of brevity, we write them in compact form as follows.

Theorem IV.43 (A D_{r} cubic sum). Let x_{1}, …, x_{r}, c_{1}, …, c_{r}, and a be indeterminate, let r ≥ 1, and suppose that none of the denominators in (IV.44) vanish. Then,

∑_{k_{i}≥0} ; i=1,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{3k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{3k_{i}−3k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(x_{i}x_{j}; q^{3})_{k_{i}+k_{j}}^{−1}

× ∏_{i,j=1}^{r} \frac{(c_{j}x_{i}/x_{j}; q^{3})_{k_{i}} (x_{i}x_{j}/c_{j}; q^{3})_{k_{i}}}{(q^{3}x_{i}/x_{j}; q^{3})_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/x_{i}; q)_{|k|−3k_{i}}}{(ax_{i}q/c_{i}; q)_{|k|} (ac_{i}q/x_{i}; q)_{|k|}}

× \frac{(1/a^{2}; q)_{|k|} (a^{2}q; q)_{2|k|}}{∏_{i=1}^{r}(a^{3}x_{i}q^{3}; q^{3})_{k_{i}}} a^{2|k|} ∏_{i=1}^{r} x_{i}^{−2k_{i}} · q^{6e_{2}(k)−\binom{2|k|+1}{2}+3 ∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{∞} (x_{i}/ac_{i}; q)_{∞} (x_{i}/a^{3}; q^{3})_{∞} (a^{3}x_{i}q^{3}/c_{i}; q^{3})_{∞}}{(ax_{i}q/c_{i}; q)_{∞} (x_{i}/a; q)_{∞} (x_{i}/a^{3}c_{i}; q^{3})_{∞} (a^{3}x_{i}q^{3}; q^{3})_{∞}} , (IV.44)

provided the series terminates (and where e_{2}(k) is the second elementary symmetric function of {k_{1}, …, k_{r}}).

Remark IV.45. The aforementioned D_{r} cubic summation theorems are the cases a^{2} = q^{N}, with N being an arbitrary integer, of Theorem IV.43, where the right side of (IV.44) simplifies differently, depending on the sign of N and the residue class of N mod 6.

CHAPTER V C_{r} and D_{r} very-well-poised _{10}ϕ_{9} transformations

Here we show how one of our main results which we obtained by matrix inversion, the D_{r} _{8}ϕ_{7} summation, Theorem IV.22, can be combined with other multidimensional results to obtain further, more complicated, identities. This chapter and the following one are part of a joint paper with G. Bhatnagar [14].

The Rogers–Ramanujan identities [1], [2], [3], [5], [6] are perhaps the most celebrated identities involving basic hypergeometric series. G. N. Watson [73] proved these identities by first finding a q-analogue of Whipple’s [74] transformation formula, taking suitable limiting cases, and then invoking Jacobi’s Triple Product Identity [41], [3]. Watson’s proof of his transformation formula was analogous to Whipple’s [75] second proof of his identity. He used a polynomial argument and Jackson’s [39] q-Pfaff–Saalschütz theorem, a summation theorem for a terminating and balanced _{3}ϕ_{2} series. Bailey [7] observed that Whipple’s ideas could be extended further, by using Jackson’s [40] q-Dougall summation theorem, which is a summation theorem for a terminating and balanced_{8}ϕ_{7} sum (see Theorem IV.8), and which contains the q-Pfaff–Saalschütz summation as a special case. The result was Bailey’s _{10}ϕ_{9} transformation formula, which converts a terminating, balanced, and very-well-poised _{10}ϕ_{9} series into a multiple of a series of the same kind.

In this chapter, we generalize Bailey’s _{10}ϕ_{9} transformation formula to multiple series. As ingredients we use A_{r}, C_{r}, and D_{r} extensions of Jackson’s [40] _{8}ϕ_{7} summation which are stated in Appendix B, and an interchange of multisums. As special cases of our _{10}ϕ_{9} transformations, we find several multivariable generalizations of Watson’s [73] transformation of an _{8}ϕ_{7} into a multiple of a _{4}ϕ_{3} (see chapter VI). We also deduce multidimensional extensions of Sears’ [69] _{4}ϕ_{3} transformation (which are also given in chapter VI).

46 V. C_{r} AND D_{r} VERY-WELL-POISED _{10}ϕ_{9} TRANSFORMATIONS

Previously, Milne and Newcomb [59] have generalized Bailey’s formula to U(r + 1) multiple series, or equivalently to A_{r} series. As special cases, they also recover many A_{r} generalizations of Watson’s transformation found earlier by Milne [54], [56]. We use similar techniques and find C_{r} and D_{r} generalizations of the _{10}ϕ_{9} transformation formula. Several different generalizations of Watson’s transformation follow from any one multivariable extension of Bailey’s formula. Thus we are able to unify all the previously known A_{r}, C_{r}, and D_{r} generalizations of Watson’s transformations [12], [52], [54], [56], [58]. In addition, we find 14 new A_{r} and D_{r} generalizations of Watson’s transformation.

The proofs of our results are generalizations of Bailey’s [8], [10] second proof of his transformation formula. Following the distillation of Bailey’s ideas by Gasper and Rahman [25], we use a suitable multivariable extension of the elementary interchange of summation argument:

∑_{m=0}^{n} ∑_{k=0}^{m}A(m) C(k, m) = ∑_{k=0}^{n} ∑_{m=0}^{n−k}A(m + k) C(k, m + k).  (V.1)

When A and C are chosen so that the inner sums are both summable using Jackson’s q-Dougall sum, we obtain the_{10}ϕ_{9} transformation formula. Essentially, Milne and Newcomb [59] use the same model in their derivation, except that they follow the organization of Bailey’s ideas in Slater’s [71] book, rather than the simpler exposition by Gasper and Rahman [25].

Milne and Newcomb [59] used Milne’s [52] A_{r} generalization of Jackson’s_{8}ϕ_{7} sum to sum up the inner sum on both sides of a suitable multivariable generalization of (V.1). However, this is not the only multivariable generalization of Jackson’s sum. Denis and Gustafson [19] and Milne and Lilly [58] independently have found a C_{r} extension of Jackson’s sum, and more recently, Bhatnagar [12] and the author [67] have found D_{r} extensions of Jackson’s sum. (The author [68] has also found a new C_{r} extension of Jackson’s sum, see Theorem VII.31 in this thesis, however it cannot be used to derive another multiple _{10}ϕ_{9} transformation, see Remark VII.33.) To derive our C_{r} and D_{r} _{10}ϕ_{9} transformations, we have used all possible combinations of these A_{r}, C_{r} and D_{r} summations (which are listed in Appendix B). As one strange consequence of combining C_{r} and D_{r} theorems, we find a generalization of Bailey’s transformation, which converts a C_{r} _{10}ϕ_{9} series into a multiple of an A_{r} 10ϕ_{9} series. Indeed, in [14] and here, while we follow the convention set by previous authors [31], [58], [12] of labelling our formulas as A_{r}, C_{r}, or D_{r} theorems, we do not hesitate to mix the different types of series if necessary.

To motivate our multidimensional work in this chapter we state the classical (one-variable)_{10}ϕ_{9} transformation [25, (2.9.1)] and give a proof, basically following the proof displayed in [25, sec. 2.9]. We will extend this analysis in the sequel of this chapter to prove our multivariable _{10}ϕ_{9} transformations of type C_{r} and D_{r}, respectively.

Theorem V.2 (Bailey’s classical _{10}ϕ_{9} transformation). Let a, b, c, d, e, and f be indeterminate, let n be a nonnegative integer, and suppose that none of the denominators in (V.3) vanish. Then

10ϕ_{9}[\begin{matrix}a, q\sqrt{a}, −q\sqrt{a}, b, c, d, e, f, λaq^{n+1}/ef, q^{−n} \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e, aq/f, efq^{−n}/λ, aq^{n+1} \end{matrix}; q, q]

= \frac{(aq; q)_{n} (aq/ef; q)_{n} (λq/e; q)_{n} (λq/f; q)_{n}}{(λq/ef; q)_{n} (λq; q)_{n} (aq/f; q)_{n} (aq/e; q)_{n}}

×_{10}ϕ_{9}[\begin{matrix}λ, q\sqrt{λ}, −q\sqrt{λ}, λb/a, λc/a, λd/a, e, f, λaq^{n+1}/ef, q^{−n} \\ \sqrt{λ}, −\sqrt{λ}, aq/b, aq/c, aq/d, λq/e, λq/f, efq^{−n}/a, λq^{n+1} \end{matrix}; q, q] , (V.3)

where λ = qa^{2}/bcd.

Proof of Theorem V.2. To derive (V.3), we start by writing the sum on the left hand side of (V.1) in the form

∑_{0≤m≤n} A(m) B(m),  (V.4)

where

A(m) = \frac{(1 − aq^{2m})}{(1 − a)} \frac{(a; q)_{m} (e; q)_{m} (f; q)_{m}}{(q; q)_{m} (aq/e; q)_{m} (aq/f; q)_{m}}

× \frac{(aλq^{1+n}/ef; q)_{m} (q^{−n}; q)_{m} (a/λ; q)_{m}}{(efq^{−n}/λ; q)_{m} (aq^{1+n}; q)_{m} (λq; q)_{m}} q^{m}

and

B(m) = \frac{(b; q)_{m} (c; q)_{m} (d; q)_{m} (λq; q)_{m}}{(aq/b; q)_{m} (aq/c; q)_{m} (aq/d; q)_{m} (a/λ; q)_{m}} .

By Theorem IV.8 we may substitute

∑_{0≤k≤m} \frac{(1 − λq^{2k})}{(1 − λ)} \frac{(λ; q)_{k} (λb/a; q)_{k} (λc/a; q)_{k}}{(q; q)_{k} (aq/b; q)_{k} (aq/c; q)_{k}}

× \frac{(λd/a; q)_{k} (aq^{m}; q)_{k} (q^{−m}; q)_{k}}{(aq/d; q)_{k} (λq^{1−m}/a; q)_{k} (λq^{1+m}; q)_{k}} q^{k} (V.5)

for B(m) in (V.4). Write C(k, m) for the summand in (V.5). Thus, the left side of (V.3) equals

∑_{0≤m≤n} ∑_{0≤k≤m} A(m) C(k, m).

Now we interchange the summations

∑_{0≤k≤n} ∑_{k≤m≤n} A(m) C(k, m),

and after shifting m to m + k in the inner sum while changing the range of summation we obtain

∑_{0≤k≤n} ∑_{0≤m≤n−k} A(m + k) C(k, m + k).  (V.6)

Explicitly, after rearranging terms, (V.6) is

∑_{0≤k≤n} \frac{(1 − λq^{2k})}{(1 − λ)} \frac{(λ; q)_{k} (λb/a; q)_{k} (λc/a; q)_{k} (λd/a; q)_{k} (aq^{k}; q)_{k} (q^{−k}; q)_{k}}{(q; q)_{k} (aq/b; q)_{k} (aq/c; q)_{k} (aq/d; q)_{k} (λq^{1−k}/a; q)_{k} (λq^{1+k}; q)_{k}} q^{k}

× \frac{(1 − aq^{2k})}{(1 − a)} \frac{(a; q)_{k} (e; q)_{k} (f; q)_{k} (a/λ; q)_{k} (aλq^{1+n}/ef; q)_{k} (q^{−n}; q)_{k}}{(q; q)_{k} (aq/e; q)_{k} (aq/f; q)_{k} (λq; q)_{k} (efq^{−n}/λ; q)_{k} (aq^{1+n}; q)_{k}} q^{k}

× ∑_{0≤m≤n−k} \frac{(1 − aq^{2k+2m})}{(1 − aq^{2k})} \frac{(aq^{2k}; q)_{m} (eq^{k}; q)_{m} (fq^{k}; q)_{m}}{(q; q)_{m} (aq^{1+k}/e; q)_{m} (aq^{1+k}/f; q)_{m}}

× \frac{(a/λ; q)_{m} (aλq^{1+k+n}/ef; q)_{m} (q^{k−n}; q)_{m}}{(λq^{1+2k}; q)_{m} (efq^{k−n}/λ; q)_{m} (aq^{1+k+n}; q)_{m}} q^{m}.

Summing the inner sum by means of the a ↦ aq^{k}, b ↦ eq^{k}, c ↦ fq^{k}, d ↦ a/λ, and n ↦ n − k case of Theorem IV.8 and simplifying terms by some elementary manipulations (cf. [25, Appendix I]), we obtain the right side of (V.3).

Remark V.7. Note that the choice of λ is motivated by our desire to sum the inner sum of (V.6) using (IV.9). In our choice of B(m), we have to choose one of our products in the numerator as corresponding to the product (aq; q)_{n} in the product side of (IV.9). So we start by choosing λ to correspond to a. Next we try to keep the other three parameters b, c, and d, as general as possible. Now forcing B(m) to fit the product side of (IV.9), leads us to λ = qa^{2}/bcd.

Notice also that the product side of (IV.9) satisfies the well-poised condition. That is,

(aq)(aq/bcd) = (aq/bc)(aq/d) = (aq/bd)(aq/c) = (aq/cd)(aq/b) = q^{2}a^{2}/bcd.

So if any of these products survive when we use (IV.9) on the right hand side of (V.6), the “special parameter” is forced to be qa^{2}/bcd. This is another justification for the choice of λ.

The choice of A(m) (and of course also of B(m)) is motivated by our desire to sum the inner sum by some summation theorem which is as general as possible (i.e. Jackson’s sum).

We proceed to higher-dimensional _{10}ϕ_{9} transformations. First, we present a C_{r} generalization of Theorem V.2. We use a straightforward multiple series extension of (V.1), and two multiple series generalizations of Theorem IV.8. These are: The D_{r} extension of Jackson’s sum derived in chapter IV and a C_{r} summation theorem found independently by Denis and Gustafson [19] and Milne and Lilly [58]. Both results are listed (for completeness and better reference) in the appendix, see Theorem B.12 and Theorem B.6.

Consider the substitutions a ↦ qa^{2}/bcd, b ↦ aq/cd, c ↦ aq/bd, d ↦ aq/bc in Theorem V.2. With these substitutions, λ becomes a, and the _{10}ϕ_{9} on the right hand side of (V.3) gets transformed into the one on the left, and vice-versa. Once again, we obtain Bailey’s formula (V.3). This is quite a striking symmetry of Bailey’s transformation. This symmetry holds for quite all the multiple series extensions of (V.3) in [59] and in this chapter. There is only one exception: Remarkably enough, it does not hold in the following generalization of Theorem V.2:

Theorem V.8 (A C_{r} _{10}ϕ_{9} transformation). Let x_{1}, …, x_{r}, a, b, c, d, e, and f be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (V.9) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q^{1+n_{j}}; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}} (dx_{i}; q)_{k_{i}}}{(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ex_{i}; q)_{k_{i}} (fx_{i}; q)_{k_{i}} (aλx_{i}q^{1+|n|}/ef; q)_{k_{i}}}{(ax_{i}q/e; q)_{k_{i}} (ax_{i}q/f; q)_{k_{i}} (efx_{i}q^{−|n|}/λ; q)_{k_{i}}} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{1≤i<j≤r}(ax_{i}x_{j}q; q)_{n_{i}+n_{j}}^{−1} ∏_{i,j=1}^{r}(ax_{i}x_{j}q; q)_{n_{i}}

× \frac{(λq/e; q)_{|n|} (λq/f; q)_{|n|} (aq/ef; q)_{|n|}}{∏_{i=1}^{r} [(λx_{i}q; q)_{n_{i}} (ax_{i}q/e; q)_{n_{i}} (ax_{i}q/f; q)_{n_{i}} (λq^{1+|n|−n_{i}}/efx_{i}; q)_{n_{i}}]}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|}}{(λx_{i}q^{1+n_{i}}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(ex_{i}; q)_{k_{i}} (fx_{i}; q)_{k_{i}} (aλx_{i}q^{1+|n|}/ef; q)_{k_{i}}}{(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}}}

× \frac{(λb/a; q)_{|k|} (λc/a; q)_{|k|} (λd/a; q)_{|k|}}{(λq/e; q)_{|k|} (λq/f; q)_{|k|} (efq^{−|n|}/a; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (V.9)

where λ = qa^{2}/bcd.

Remark V.10. We call Theorem V.8 a C_{r} theorem, since a limiting case yields Milne and Lilly’s [58] C_{r} generalization of Watson’s transformation. However, note that (V.9) transforms a terminating, balanced, and very-well-poised C_{r} _{10}ϕ_{9} sum into a multiple of a terminating A_{r} series, which is also balanced and very-well-poised. In view of our earlier remarks, it may also be written so that it transforms an A_{r} series into a C_{r} series.

Proof. The left side of (V.9) can be written in the form

∑_{0≤m_{i}≤n_{i}} ; i=1,2,...,rA(m) B(m),  (V.11)

where

A(m) = ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2m_{i}}}{1 − ax_{i}^{2}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{m_{i}−m_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} \frac{1 − ax_{i}x_{j}q^{m_{i}+m_{j}}}{1 − ax_{i}x_{j}} )

× ∏_{i=1}^{r} \frac{(ex_{i}; q)_{m_{i}} (fx_{i}; q)_{m_{i}} (aλx_{i}q^{1+|n|}/ef; q)_{m_{i}} (ax_{i}/λ; q)_{m_{i}}}{(ax_{i}q/e; q)_{m_{i}} (ax_{i}q/f; q)_{m_{i}} (efx_{i}q^{−|n|}/λ; q)_{m_{i}} (λx_{i}q; q)_{m_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{m_{i}} (ax_{i}x_{j}; q)_{m_{i}}}{(qx_{i}/x_{j}; q)_{m_{i}} (ax_{i}x_{j}q^{1+n_{j}}; q)_{m_{i}}} q^{∑_{i=1}^{r} i m_{i}}

and

B(m) = ∏_{i=1}^{r} \frac{(bx_{i}; q)_{m_{i}} (cx_{i}; q)_{m_{i}} (dx_{i}; q)_{m_{i}} (λx_{i}q; q)_{m_{i}}}{(ax_{i}q/b; q)_{m_{i}} (ax_{i}q/c; q)_{m_{i}} (ax_{i}q/d; q)_{m_{i}} (ax_{i}/λ; q)_{m_{i}}} .

By Theorem B.12 we may substitute

∑_{0≤k_{i}≤m_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}; q)_{k_{i}+k_{j}}^{−1} ∏_{i,j=1}^{r}(ax_{i}x_{j}q^{m_{j}}; q)_{k_{i}}

× ∏_{i,j=1}^{r} \frac{(q^{−m_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λq/ax_{i}; q)_{|k|−k_{i}}}{(λx_{i}q^{1+m_{i}}; q)_{|k|} (λq^{1−m_{i}}/ax_{i}; q)_{|k|}}

× \frac{(λb/a; q)_{|k|} (λc/a; q)_{|k|} (λd/a; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}}]} q^{∑_{i=1}^{r} i k_{i}} ) (V.12)

for B(m) in (V.11). Write C(k, m) for the summand in (V.12). Thus, the left side of (V.9) equals

∑_{0≤m_{i}≤n_{i}} ; i=1,...,r _{∑_{0≤k_{i}≤m_{i}} ; i=1,...,r}A(m) C(k, m).

Now we interchange the summations

∑_{0≤k_{i}≤n_{i}} ; i=1,...,r _{∑_{k_{i}≤m_{i}≤n_{i}} ; i=1,...,r}A(m) C(k, m),

and after shifting each m_{i} to m_{i} + k_{i} in the inner sum while changing the range of summation we obtain

∑_{0≤k_{i}≤n_{i}} ; i=1,...,r _{∑_{0≤m_{i}≤n_{i}−k_{i}} ; i=1,...,r}A(m + k) C(k, m + k).  (V.13)

Explicitly, after rearranging terms, (V.13) is

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}; q)_{k_{i}+k_{j}}^{−1} ∏_{i,j=1}^{r}(ax_{i}x_{j}q^{k_{j}}; q)_{k_{i}}

× ∏_{i,j=1}^{r} \frac{(q^{−k_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λq/ax_{i}; q)_{|k|−k_{i}}}{(λx_{i}q^{1+k_{i}}; q)_{|k|} (λq^{1−k_{i}}/ax_{i}; q)_{|k|}}

× \frac{(λb/a; q)_{|k|} (λc/a; q)_{|k|} (λd/a; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}}]} q^{∑_{i=1}^{r} i k_{i}}

× ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} )

× ∏_{i=1}^{r} \frac{(ex_{i}; q)_{k_{i}} (fx_{i}; q)_{k_{i}} (aλx_{i}q^{1+|n|}/ef; q)_{k_{i}} (ax_{i}/λ; q)_{k_{i}}}{(ax_{i}q/e; q)_{k_{i}} (ax_{i}q/f; q)_{k_{i}} (efx_{i}q^{−|n|}/λ; q)_{k_{i}} (λx_{i}q; q)_{k_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q^{1+n_{j}}; q)_{k_{i}}} q^{∑_{i=1}^{r} i k_{i}}

×_{∑_{0≤m_{i}≤n_{i}−k_{i}} ; i=1,2,...,r} ∏_{1≤i<j≤r}(\frac{1 − q^{m_{i}−m_{j}+k_{i}−k_{j}}x_{i}/x_{j}}{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}+m_{i}+m_{j}}}{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}} )

× ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}+2m_{i}}}{1 − ax_{i}^{2}q^{2k_{i}}} ) ∏_{i=1}^{r} \frac{(ex_{i}q^{k_{i}}; q)_{m_{i}} (fx_{i}q^{k_{i}}; q)_{m_{i}}}{(ax_{i}q^{1+k_{i}}/e; q)_{m_{i}} (ax_{i}q^{1+k_{i}}/f; q)_{m_{i}}}

× ∏_{i=1}^{r} \frac{(aλx_{i}q^{1+k_{i}+|n|}/ef; q)_{m_{i}} (ax_{i}q^{k_{i}−|k|}/λ; q)_{m_{i}}}{(efx_{i}q^{k_{i}−|n|}/λ; q)_{m_{i}} (λx_{i}q^{1+k_{i}+|k|}; q)_{m_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{k_{i}−n_{j}}x_{i}/x_{j}; q)_{m_{i}} (ax_{i}x_{j}q^{k_{i}+k_{j}}; q)_{m_{i}}}{(q^{1+k_{i}−k_{j}}x_{i}/x_{j}; q)_{m_{i}} (ax_{i}x_{j}q^{1+k_{i}+n_{j}}; q)_{m_{i}}} q^{∑_{i=1}^{r} i m_{i}} ).

Summing the inner sum by means of the

b ↦ e,  c ↦ f,  d ↦ aq^{−|k|}/λ,
x_{i} ↦ x_{i}q^{k_{i}}, n_{i} ↦ n_{i} − k_{i}, for i = 1, 2, …, r

case of Theorem B.6 and simplifying terms by the l_{1} = l_{2} = ⋯ = l_{r} = 0 case of Lemma B.1 and some elementary manipulations (cf. [25, Appendix I]), we obtain the right side of (V.9).

Remark V.14. An alternative way to derive Theorem V.8 is to use Denis and Gustafson’s, respectively Milne and Lilly’s, C_{r} _{8}ϕ_{7} summation, Theorem B.6, in the inner sum, and then employ Bhatnagar’s D_{r} _{8}ϕ_{7} summation, Theorem B.9, after exchanging sums. Then we would arrive at identity V.9 too, but with the right hand side on the left and vice-versa.

We will discuss some special cases of our C_{r} _{10}ϕ_{9} transformation in chapter VI. Some more multiple _{10}ϕ_{9} transformations can be derived by the above method.

If we combine both D_{r} _{8}ϕ_{7} summations, Theorem B.9 and Theorem B.12 we obtain the following D_{r} _{10}ϕ_{9} transformation formula.

Theorem V.15 (A D_{r} _{10}ϕ_{9} transformation). Let x_{1}, …, x_{r}, a, b, c, d, e, and f be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (V.16) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} \frac{(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}}}{(λax_{i}x_{j}q/ef; q)_{k_{i}+k_{j}}} ∏_{i=1}^{r} \frac{(ef/λx_{i}; q)_{|k|−k_{i}}}{(d/x_{i}; q)_{|k|−k_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (λax_{i}x_{j}q^{1+n_{j}}/ef; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/d; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (d/x_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (efq^{−n_{i}}/λx_{i}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}}}{(ax_{i}q/e; q)_{k_{i}} (ax_{i}q/f; q)_{k_{i}}}

× \frac{(e; q)_{|k|} (f; q)_{|k|}}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/ef; q)_{n_{i}} (λx_{i}q/e; q)_{n_{i}} (λx_{i}q/f; q)_{n_{i}}}{(λx_{i}q/ef; q)_{n_{i}} (λx_{i}q; q)_{n_{i}} (ax_{i}q/f; q)_{n_{i}} (ax_{i}q/e; q)_{n_{i}}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} \frac{(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}}}{(λax_{i}x_{j}q/ef; q)_{k_{i}+k_{j}}} ∏_{i=1}^{r} \frac{(ef/ax_{i}; q)_{|k|−k_{i}}}{(λd/ax_{i}; q)_{|k|−k_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (λax_{i}x_{j}q^{1+n_{j}}/ef; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/d; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λd/ax_{i}; q)_{|k|}}{(λx_{i}q^{1+n_{i}}; q)_{|k|} (efq^{−n_{i}}/ax_{i}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(λbx_{i}/a; q)_{k_{i}} (λcx_{i}/a; q)_{k_{i}}}{(λx_{i}q/e; q)_{k_{i}} (λx_{i}q/f; q)_{k_{i}}}

× \frac{(e; q)_{|k|} (f; q)_{|k|}}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (V.16)

where λ = qa^{2}/bcd. Proof. The left side of (V.16) can be written in the form (V.11) where

A(m) = ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{m_{i}+|m|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{m_{i}−m_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(λax_{i}x_{j}q/ef; q)_{m_{i}+m_{j}}^{−1} ∏_{i,j=1}^{r}(λax_{i}x_{j}q^{1+n_{j}}/ef; q)_{m_{i}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{m_{i}}}{(qx_{i}/x_{j}; q)_{m_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|m|} (ef/λx_{i}; q)_{|m|−m_{i}}}{(ax_{i}q^{1+n_{i}}; q)_{|m|} (efq^{−n_{i}}/λx_{i}; q)_{|m|}}

54 V. C_{r} AND D_{r} VERY-WELL-POISED _{10}ϕ_{9} TRANSFORMATIONS

× \frac{(e; q)_{|m|} (f; q)_{|m|} (a/λ; q)_{|m|}}{∏_{i=1}^{r} [(ax_{i}q/e; q)_{m_{i}} (ax_{i}q/f; q)_{m_{i}} (λx_{i}q; q)_{m_{i}}]} q^{∑_{i=1}^{r} i m_{i}}

and

B(m) = ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{m_{i}+m_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{m_{i}}^{−1}

× \frac{∏_{i=1}^{r} [(bx_{i}; q)_{m_{i}} (cx_{i}; q)_{m_{i}} (dq^{|m|−m_{i}}/x_{i}; q)_{m_{i}} (λx_{i}q; q)_{m_{i}}]}{(aq/b; q)_{|m|} (aq/c; q)_{|m|} (a/λ; q)_{|m|}}.

By Theorem B.9 we may substitute

∑_{0≤k_{i}≤m_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{k_{i}}^{−1}

× ∏_{i,j=1}^{r} \frac{(q^{−m_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λd/ax_{i}; q)_{|k|}}{(λx_{i}q^{1+m_{i}}; q)_{|k|} (λd/ax_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(λbx_{i}/a; q)_{k_{i}} (λcx_{i}/a; q)_{k_{i}} (ax_{i}q^{|m|}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (λq^{1−|m|}/a; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ) (V.17)

for B(m) in (V.11). Write C(k, m) for the summand in (V.17). As in the proof of Theorem V.8, the left side of (V.16) equals

∑_{0≤k_{i}≤n_{i}} ; i=1,...,r _{∑_{0≤m_{i}≤n_{i}−k_{i}} ; i=1,...,r}A(m + k) C(k, m + k).  (V.18)

Explicitly, after rearranging terms, (V.18) is

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{k_{i}}^{−1}

× ∏_{i,j=1}^{r} \frac{(q^{−k_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λd/ax_{i}; q)_{|k|}}{(λx_{i}q^{1+k_{i}}; q)_{|k|} (λd/ax_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(λbx_{i}/a; q)_{k_{i}} (λcx_{i}/a; q)_{k_{i}} (ax_{i}q^{|k|}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (λq^{1−|k|}/a; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}}

× ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(λax_{i}x_{j}q/ef; q)_{k_{i}+k_{j}}^{−1} ∏_{i,j=1}^{r}(λax_{i}x_{j}q^{1+n_{j}}/ef; q)_{k_{i}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (ef/λx_{i}; q)_{|k|−k_{i}}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (efq^{−n_{i}}/λx_{i}; q)_{|k|}}

× \frac{(e; q)_{|k|} (f; q)_{|k|} (a/λ; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/e; q)_{k_{i}} (ax_{i}q/f; q)_{k_{i}} (λx_{i}q; q)_{k_{i}}]} q^{∑_{i=1}^{r} i k_{i}}

×_{∑_{0≤m_{i}≤n_{i}−k_{i}} ; i=1,2,...,r} ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|+m_{i}+|m|}}{1 − ax_{i}q^{k_{i}+|k|}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{m_{i}−m_{j}+k_{i}−k_{j}}x_{i}/x_{j}}{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(λax_{i}x_{j}q^{1+k_{i}+k_{j}}/ef; q)_{m_{i}+m_{j}}^{−1} ∏_{i,j=1}^{r}(λax_{i}x_{j}q^{1+k_{i}+n_{j}}/ef; q)_{m_{i}}

× ∏_{i,j=1}^{r} \frac{(q^{k_{i}−n_{j}}x_{i}/x_{j}; q)_{m_{i}}}{(q^{1+k_{i}−k_{j}}x_{i}/x_{j}; q)_{m_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}q^{k_{i}+|k|}; q)_{|m|} (efq^{|k|−k_{i}}/λx_{i}; q)_{|m|−m_{i}}}{(ax_{i}q^{1+|k|+n_{i}}; q)_{|m|} (efq^{|k|−n_{i}}/λx_{i}; q)_{|m|}}

× \frac{(eq^{|k|}; q)_{|m|} (fq^{|k|}; q)_{|m|} (a/λ; q)_{|m|}}{∏_{i=1}^{r} [(ax_{i}q^{1+k_{i}}/e; q)_{m_{i}} (ax_{i}q^{1+k_{i}}/f; q)_{m_{i}} (λx_{i}q^{1+k_{i}+|k|}; q)_{m_{i}}]} q^{∑_{i=1}^{r} i m_{i}} ).

Summing the inner sum by means of the

a ↦ aq^{|k|}, b ↦ eq^{|k|},  c ↦ fq^{|k|}, d ↦ a/λ,
x_{i} ↦ x_{i}q^{k_{i}}, n_{i} ↦ n_{i} − k_{i}, for i = 1, 2, …, r

case of Theorem B.12 and simplifying terms by the l_{1} = l_{2} = ⋯ = l_{r} = 0 case of Lemma B.1 and some elementary manipulations, we obtain the right side of (V.16).

Remark V.19. Some special limiting cases of Theorem V.15 give multiple Watson transformations. The various possibilities of specializing for Watson’s will be analyzed in chapter VI. This remark holds for all our multiple _{10}ϕ_{9} transformations.

By using a polynomial argument we easily obtain

Theorem V.20 (A D_{r} _{10}ϕ_{9} transformation). Let x_{1}, …, x_{r}, a, b, c, d, e, and f_{1} …, f_{r} be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (V.21) vanish. Then

∑_{k_{1},k_{2},...,k_{r}≥0} ; 0≤|k|≤N( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} \frac{(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}}}{(λax_{i}x_{j}q^{1+N}/e; q)_{k_{i}+k_{j}}} ∏_{i=1}^{r} \frac{(eq^{−N}/λx_{i}; q)_{|k|−k_{i}}}{(d/x_{i}; q)_{|k|−k_{i}}}

× ∏_{i,j=1}^{r} \frac{(f_{j}x_{i}/x_{j}; q)_{k_{i}} (λax_{i}x_{j}q^{1+N}/ef_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/d; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (d/x_{i}; q)_{|k|}}{(ax_{i}q/f_{i}; q)_{|k|} (ef_{i}q^{−N}/λx_{i}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}}}{(ax_{i}q/e; q)_{k_{i}} (ax_{i}q^{1+N}; q)_{k_{i}}}

× \frac{(e; q)_{|k|} (q^{−N}; q)_{|k|}}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{N} (ax_{i}q/ef_{i}; q)_{N} (λx_{i}q/e; q)_{N} (λx_{i}q/f_{i}; q)_{N}}{(λx_{i}q/ef_{i}; q)_{N} (λx_{i}q; q)_{N} (ax_{i}q/f_{i}; q)_{N} (ax_{i}q/e; q)_{N}}

×_{∑_{k_{1},k_{2},...,k_{r}≥0} ; 0≤|k|≤N}( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} \frac{(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}}}{(λax_{i}x_{j}q^{1+N}/e; q)_{k_{i}+k_{j}}} ∏_{i=1}^{r} \frac{(eq^{−N}/ax_{i}; q)_{|k|−k_{i}}}{(λd/ax_{i}; q)_{|k|−k_{i}}}

× ∏_{i,j=1}^{r} \frac{(f_{j}x_{i}/x_{j}; q)_{k_{i}} (λax_{i}x_{j}q^{1+N}/ef_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/d; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λd/ax_{i}; q)_{|k|}}{(λx_{i}q/f_{i}; q)_{|k|} (ef_{i}q^{−N}/ax_{i}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(λbx_{i}/a; q)_{k_{i}} (λcx_{i}/a; q)_{k_{i}}}{(λx_{i}q/e; q)_{k_{i}} (λx_{i}q^{1+N}; q)_{k_{i}}}

× \frac{(e; q)_{|k|} (q^{−N}; q)_{|k|}}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (V.21)

where λ = qa^{2}/bcd.

Proof. First we write the product in front of the sum of the right side of (V.21) as quotient of infinite products using (IV.3). Then by the f = q^{−N} case of Theorem V.15 it follows that the identity (V.21) holds for f_{j} = q^{−n_{j}}, j = 1, …, r. By clearing out denominators in (V.21), we get a polynomial equation in f_{1}, which is true for q^{−n_{1}}, n_{1} = 0, 1, . . .. Thus we obtain an identity in f_{1}. By carrying out this process for f_{2}, f_{3}, …, f_{r} also, we obtain Theorem V.20.

Instead of using Theorem B.12 in the derivation of Theorem V.15, we may also use Milne’s A_{r} _{8}ϕ_{7} summation, Theorem B.3, to get another _{10}ϕ_{9} transformation.

Theorem V.22 (A D_{r} _{10}ϕ_{9} transformation). Let x_{1}, …, x_{r}, a, b, c, d, e, and f be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (V.23) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{k_{i}}^{−1}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (d/x_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (d/x_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}} (ex_{i}; q)_{k_{i}} (λax_{i}q^{1+|n|}/ef; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (aq/e; q)_{|k|} (efq^{−|n|}/λ; q)_{|k|}}

× \frac{(f; q)_{|k|}}{∏_{i=1}^{r}(ax_{i}q/f; q)_{k_{i}}} q^{∑_{i=1}^{r} i k_{i}} )

= \frac{(aq/ef; q)_{|n|} (λq/e; q)_{|n|}}{(λq/ef; q)_{|n|} (aq/e; q)_{|n|}} ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{n_{i}} (λx_{i}q/f; q)_{n_{i}}}{(λx_{i}q; q)_{n_{i}} (ax_{i}q/f; q)_{n_{i}}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{k_{i}}^{−1}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λd/ax_{i}; q)_{|k|}}{(λx_{i}q^{1+n_{i}}; q)_{|k|} (λd/ax_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(λbx_{i}/a; q)_{k_{i}} (λcx_{i}/a; q)_{k_{i}} (ex_{i}; q)_{k_{i}} (λax_{i}q^{1+|n|}/ef; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (λq/e; q)_{|k|} (efq^{−|n|}/a; q)_{|k|}}

× \frac{(f; q)_{|k|}}{∏_{i=1}^{r}(λx_{i}q/f; q)_{k_{i}}} q^{∑_{i=1}^{r} i k_{i}} ), (V.23)

where λ = qa^{2}/bcd.

Proof. The left side of (V.23) can be written in the form (V.11) where

A(m) = ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{m_{i}+|m|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{m_{i}−m_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{m_{i}}}{(qx_{i}/x_{j}; q)_{m_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|m|}}{(ax_{i}q^{1+n_{i}}; q)_{|m|}}

× ∏_{i=1}^{r} \frac{(ex_{i}; q)_{m_{i}} (λax_{i}q^{1+|n|}/ef; q)_{m_{i}}}{(ax_{i}q/f; q)_{m_{i}} (λx_{i}q; q)_{m_{i}}}

× \frac{(f; q)_{|m|} (a/λ; q)_{|m|}}{(aq/e; q)_{|m|} (efq^{−|n|}/λ; q)_{|m|}} q^{∑_{i=1}^{r} i m_{i}}

and

B(m) = ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{m_{i}+m_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{m_{i}}^{−1}

× \frac{∏_{i=1}^{r} [(bx_{i}; q)_{m_{i}} (cx_{i}; q)_{m_{i}} (dq^{|m|−m_{i}}/x_{i}; q)_{m_{i}} (λx_{i}q; q)_{m_{i}}]}{(aq/b; q)_{|m|} (aq/c; q)_{|m|} (a/λ; q)_{|m|}}.

By Theorem B.9 we may substitute (V.17) for B(m) in (V.11). Write C(k, m) for the summand in (V.17). As in the proofs of Theorems V.8 and V.15, the left side of (V.23) equals

∑_{0≤k_{i}≤n_{i}} ; i=1,...,r _{∑_{0≤m_{i}≤n_{i}−k_{i}} ; i=1,...,r}A(m + k) C(k, m + k).

Explicitly, after rearranging terms, this is

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{k_{i}}^{−1}

× ∏_{i,j=1}^{r} \frac{(q^{−k_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λd/ax_{i}; q)_{|k|}}{(λx_{i}q^{1+k_{i}}; q)_{|k|} (λd/ax_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(λbx_{i}/a; q)_{k_{i}} (λcx_{i}/a; q)_{k_{i}} (ax_{i}q^{|k|}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (λq^{1−|k|}/a; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}}

× ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(ex_{i}; q)_{k_{i}} (λax_{i}q^{1+|n|}/ef; q)_{k_{i}}}{(ax_{i}q/f; q)_{k_{i}} (λx_{i}q; q)_{k_{i}}}

× \frac{(f; q)_{|k|} (a/λ; q)_{|k|}}{(aq/e; q)_{|k|} (efq^{−|n|}/λ; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}}

×_{∑_{0≤m_{i}≤n_{i}−k_{i}} ; i=1,2,...,r} ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|+m_{i}+|m|}}{1 − ax_{i}q^{k_{i}+|k|}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{m_{i}−m_{j}+k_{i}−k_{j}}x_{i}/x_{j}}{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{k_{i}−n_{j}}x_{i}/x_{j}; q)_{m_{i}}}{(q^{1+k_{i}−k_{j}}x_{i}/x_{j}; q)_{m_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}q^{k_{i}+|k|}; q)_{|m|}}{(ax_{i}q^{1+|k|+n_{i}}; q)_{|m|}}

× ∏_{i=1}^{r} \frac{(ex_{i}q^{k_{i}}; q)_{m_{i}} (λax_{i}q^{1+k_{i}+|n|}/ef; q)_{m_{i}}}{(ax_{i}q^{1+k_{i}}/f; q)_{m_{i}} (λx_{i}q^{1+k_{i}+|k|}; q)_{m_{i}}}

× \frac{(fq^{|k|}; q)_{|m|} (a/λ; q)_{|m|}}{(aq^{1+|k|}/e; q)_{|m|} (efq^{|k|−|n|}/λ; q)_{|m|}} q^{∑_{i=1}^{r} i m_{i}} ).

Summing the inner sum by means of the

a ↦ aq^{|k|}, b ↦ fq^{|k|}, c ↦ a/λ, d ↦ e,
x_{i} ↦ x_{i}q^{k_{i}}, n_{i} ↦ n_{i} − k_{i}, for i = 1, 2, …, r

case of Theorem B.3 and simplifying terms by the l_{1} = l_{2} = ⋯ = l_{r} = 0 case of Lemma B.1 and some elementary manipulations, we obtain the right side of (V.23).

By using a polynomial argument we obtain

Theorem V.24 (A D_{r} _{10}ϕ_{9} transformation). Let x_{1}, …, x_{r}, a, b, c, d, e, and f_{1} …, f_{r} be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (V.25) vanish. Then

∑_{k_{1},k_{2},...,k_{r}≥0} ; 0≤|k|≤N( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{k_{i}}^{−1}

× ∏_{i,j=1}^{r} \frac{(f_{j}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (d/x_{i}; q)_{|k|}}{(ax_{i}q/f_{i}; q)_{|k|} (d/x_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}} (ex_{i}; q)_{k_{i}} (λax_{i}q^{1+N}/ef_{1} ⋯ f_{r}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (aq/e; q)_{|k|} (ef_{1} ⋯ f_{r}q^{−N}/λ; q)_{|k|}}

× \frac{(q^{−N}; q)_{|k|}}{∏_{i=1}^{r}(ax_{i}q^{1+N}; q)_{k_{i}}} q^{∑_{i=1}^{r} i k_{i}} )

= \frac{(aq/ef_{1} ⋯ f_{r}; q)_{N} (λq/e; q)_{N}}{(λq/ef_{1} ⋯ f_{r}; q)_{N} (aq/e; q)_{N}} ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{N} (λx_{i}q/f_{i}; q)_{N}}{(λx_{i}q; q)_{N} (ax_{i}q/f_{i}; q)_{N}}

×_{∑_{k_{1},k_{2},...,k_{r}≥0} ; 0≤|k|≤N}( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{k_{i}}^{−1}

× ∏_{i,j=1}^{r} \frac{(f_{j}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (λd/ax_{i}; q)_{|k|}}{(λx_{i}q/f_{i}; q)_{|k|} (λd/ax_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(λbx_{i}/a; q)_{k_{i}} (λcx_{i}/a; q)_{k_{i}} (ex_{i}; q)_{k_{i}} (λax_{i}q^{1+N}/ef_{1} ⋯ f_{r}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (λq/e; q)_{|k|} (ef_{1} ⋯ f_{r}q^{−N}/a; q)_{|k|}}

× \frac{(q^{−N}; q)_{|k|}}{∏_{i=1}^{r}(λx_{i}q^{1+N}; q)_{k_{i}}} q^{∑_{i=1}^{r} i k_{i}} ), (V.25)

where λ = qa^{2}/bcd.

Proof. First we write the product in front of the sum of the right side of (V.25) as quotient of infinite products using (IV.3). Then by the f = q^{−N} case of Theorem V.22 it follows that the identity (V.25) holds for f_{j} = q^{−n_{j}}, j = 1, …, r. By clearing out denominators in (V.25), we get a polynomial equation in f_{1}, which is true for q^{−n_{1}}, n_{1} = 0, 1, . . .. Thus we obtain an identity in f_{1}. By carrying out this process for f_{2}, f_{3}, …, f_{r} also, we obtain Theorem V.24.

By reversing sums in Theorem V.22 we get

Theorem V.26 (A D_{r} _{10}ϕ_{9} transformation). Let x_{1}, …, x_{r}, a, b, c, d, e, and f be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (V.27) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(λax_{i}x_{j}q/ef; q)_{k_{i}+k_{j}}^{−1} ∏_{i,j=1}^{r}(λax_{i}x_{j}q^{1+n_{j}}/ef; q)_{k_{i}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (ef/λx_{i}; q)_{|k|−k_{i}}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (efq^{−n_{i}}/λx_{i}; q)_{|k|}}

× \frac{(c; q)_{|k|} (d; q)_{|k|} (e; q)_{|k|} (f; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}} (ax_{i}q/e; q)_{k_{i}} (ax_{i}q/f; q)_{k_{i}}]}

× \frac{∏_{i=1}^{r}(bx_{i}; q)_{k_{i}}}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/ef; q)_{n_{i}} (λx_{i}q/e; q)_{n_{i}} (λx_{i}q/f; q)_{n_{i}}}{(λx_{i}q/ef; q)_{n_{i}} (λx_{i}q; q)_{n_{i}} (ax_{i}q/f; q)_{n_{i}} (ax_{i}q/e; q)_{n_{i}}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(λax_{i}x_{j}q/ef; q)_{k_{i}+k_{j}}^{−1} ∏_{i,j=1}^{r}(λax_{i}x_{j}q^{1+n_{j}}/ef; q)_{k_{i}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (ef/ax_{i}; q)_{|k|−k_{i}}}{(λx_{i}q^{1+n_{i}}; q)_{|k|} (efq^{−n_{i}}/ax_{i}; q)_{|k|}}

× \frac{(λc/a; q)_{|k|} (λd/a; q)_{|k|} (e; q)_{|k|} (f; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}} (λx_{i}q/e; q)_{k_{i}} (λx_{i}q/f; q)_{k_{i}}]}

× \frac{∏_{i=1}^{r}(λbx_{i}/a; q)_{k_{i}}}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (V.27)

where λ = qa^{2}/bcd.

Proof. First, replace k_{i} by n_{i} − k_{i}, for i = 1, 2, …, r, on both sides of Theorem V.22 and simplify terms by the l_{1} = l_{2} = ⋯ = l_{r} = 0 case of Lemma B.1 and some elementary manipulations (cf. [25, Appendix I]). Finally, relabel

a ↦ q^{−|n|}/a, b ↦ c/a,  c ↦ d/a,  d ↦ λq^{1−|n|}/ef,
e ↦ e/a,  f ↦ bq^{−|n|}/a, x_{i} ↦ q^{−n_{i}}/x_{i}, for i = 1, 2, …, r

in the resulting identity to obtain (V.27).

By using a polynomial argument we obtain

Theorem V.28 (A D_{r} _{10}ϕ_{9} transformation). Let x_{1}, …, x_{r}, a, b, c, d, e, and f_{1}, …, f_{r} be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (V.29) vanish. Then

∑_{k_{1},k_{2},...,k_{r}≥0} ; 0≤|k|≤N( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(λax_{i}x_{j}q^{1+N}/e; q)_{k_{i}+k_{j}}^{−1} ∏_{i,j=1}^{r}(λax_{i}x_{j}q^{1+N}/ef_{j}; q)_{k_{i}}

× ∏_{i,j=1}^{r} \frac{(f_{j}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (eq^{−N}/λx_{i}; q)_{|k|−k_{i}}}{(ax_{i}q/f_{i}; q)_{|k|} (ef_{i}q^{−N}/λx_{i}; q)_{|k|}}

× \frac{(c; q)_{|k|} (d; q)_{|k|} (e; q)_{|k|} (q^{−N}; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}} (ax_{i}q/e; q)_{k_{i}} (ax_{i}q^{1+N}; q)_{k_{i}}]}

× \frac{∏_{i=1}^{r}(bx_{i}; q)_{k_{i}}}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{N} (ax_{i}q/ef_{i}; q)_{N} (λx_{i}q/e; q)_{N} (λx_{i}q/f_{i}; q)_{N}}{(λx_{i}q/ef_{i}; q)_{N} (λx_{i}q; q)_{N} (ax_{i}q/f_{i}; q)_{N} (ax_{i}q/e; q)_{N}}

×_{∑_{k_{1},k_{2},...,k_{r}≥0} ; 0≤|k|≤N}( ∏_{i=1}^{r} (\frac{1 − λx_{i}q^{k_{i}+|k|}}{1 − λx_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(λax_{i}x_{j}q^{1+N}/e; q)_{k_{i}+k_{j}}^{−1} ∏_{i,j=1}^{r}(λax_{i}x_{j}q^{1+N}/ef_{j}; q)_{k_{i}}

× ∏_{i,j=1}^{r} \frac{(f_{j}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(λx_{i}; q)_{|k|} (eq^{−N}/ax_{i}; q)_{|k|−k_{i}}}{(λx_{i}q/f_{i}; q)_{|k|} (ef_{i}q^{−N}/ax_{i}; q)_{|k|}}

× \frac{(λc/a; q)_{|k|} (λd/a; q)_{|k|} (e; q)_{|k|} (q^{−N}; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}} (λx_{i}q/e; q)_{k_{i}} (λx_{i}q^{1+N}; q)_{k_{i}}]}

× \frac{∏_{i=1}^{r}(λbx_{i}/a; q)_{k_{i}}}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (V.29)

where λ = qa^{2}/bcd.

Proof. First we write the product in front of the sum of the right side of (V.29) as quotient of infinite products using (IV.3). Then by the f = q^{−N} case of Theorem V.26 it follows that the identity (V.29) holds for f_{j} = q^{−n_{j}}, j = 1, …, r. By clearing out denominators in (V.29), we get a polynomial equation in f_{1}, which is true for q^{−n_{1}}, n_{1} = 0, 1, . . .. Thus we obtain an identity in f_{1}. By carrying out this process for f_{2}, f_{3}, …, f_{r} also, we obtain Theorem V.28.

64 V. C_{r} AND D_{r} VERY-WELL-POISED _{10}ϕ_{9} TRANSFORMATIONS

CHAPTER VI Watson’s and Sears’ transformations

In this chapter, we work out the A_{r}, C_{r}, and D_{r} extensions of Watson’s transformation and some of the multiple Sears’ transformations which follow from our results in chapter V. As special cases of our generalizations of Bailey’s _{10}ϕ_{9} transformation, we obtain A_{r}, C_{r}, and D_{r} generalizations of Watson’s transformation found by Milne [52], [54], [56], Milne and Lilly [58], and Bhatnagar [12], respectively. See also [46], [59], [61] for some of Milne’s A_{r} generalizations of Watson’s theorem. In addition to recovering all such transformations known previously, we also obtain two new A_{r}, and several D_{r} generalizations of Watson’s theorem.

We commence this chapter by stating Watson’s classical q-Whipple transformation and our new multiple extensions thereof. Subsequently, we list some of Milne’s A_{r} Watson’s transformations, which follow immediately from our multiple _{10}ϕ_{9} transformations. We thank Professor S. C. Milne for showing us his notes for [56]. Finally, after stating a D_{r} Watson’s transformation recently found by Bhatnagar [12], we conclude this chapter with some multidimensional Sears’ transformations.

Watson’s formula follows from Bailey’s_{10}ϕ_{9} transformation in one of many ways. For instance, if we take the limit as d → ∞ in Theorem V.2, and replace f by d in the resulting identity, we obtain [25, equation (2.5.1)]:

Theorem VI.1 (Watson’s classical q-Whipple transformation). Let a, b, c, d, and e be indeterminate, let n be a nonnegative integer, and suppose that none of the denominators in (VI.2) vanish. Then

8ϕ_{7} [\begin{matrix}a, q\sqrt{a}, −q\sqrt{a}, b, c, d, e, q^{−n} \\ \sqrt{a}, −\sqrt{a}, aq/b, aq/c, aq/d, aq/e, aq^{n+1} \end{matrix}; q, \frac{a^{2}q^{2+n}}{bcde} ]

= \frac{(aq; q)_{n} (aq/de; q)_{n}}{(aq/d; q)_{n} (aq/e; q)_{n}} _{4}ϕ_{3} [\begin{matrix}aq/bc, d, e, q^{−n} \\ aq/b, aq/c, deq^{−n}/a \end{matrix}; q, q] . (VI.2)

This is not the only the only way to obtain Theorem VI.1 from (V.3). We may also take the limit b → ∞, or c → ∞. Further, we may first restate Theorem V.2 by replacing e or f by λaq^{n+1}/ef in (V.3), and then take the limits as b, c, or d → ∞, to obtain Watson’s transformation.

All the above limiting cases of Bailey’s formula lead to the same identity, up to a trivial relabelling of parameters. However, in our multivariable generalizations of Bailey’s formula, some of its symmetry is broken, and we obtain many different generalizations of Watson’s transformation. For example, while the b, c, or d → ∞ case of Theorem V.8 yields the same identity, letting b → ∞ and d → ∞ in Theorem V.15 yield different generalizations of Theorem VI.1.

Finally, as noted in chapter V we may set a ↦ qa^{2}/bcd, b ↦ aq/cd, c ↦ aq/bd, d ↦ aq/bc in Theorem V.2. With these substitutions, λ becomes a, and the _{10}ϕ_{9} on the right hand side of (V.3) gets transformed into the one on the left, and viceversa. Once again, we obtain Bailey’s _{10}ϕ_{9} formula. Now, we may take limits as above and obtain Watson’s transformation. This observation does not lead to any further generalizations of (VI.2) from our D_{r} _{10}ϕ_{9} transformations in chapter V, but it does apply to our C_{r} _{10}ϕ_{9} transformation, Theorem V.8.

If we perform the above substitutions and let d → ∞ in Theorem V.8, we obtain one of Milne’s [56] A_{r} Watson’s transformations, see [59, Theorem 5.1].

Instead, if we simply let d → ∞ in Theorem V.8, we obtain Milne and Lilly’s [58, Theorem 6.6] C_{r} generalization of Theorem VI.1.

Next we consider the multivariable generalizations of Watson’s transformations following from Theorem V.15. First we have an A_{r} extension of Theorem VI.1.

Theorem VI.3 (An A_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers for with r ≥ 1, and suppose that none of the denominators in (VI.4) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}}}{(ax_{i}q/d; q)_{k_{i}} (ax_{i}q/e; q)_{k_{i}}}

× \frac{(d; q)_{|k|} (e; q)_{|k|}}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}} (\frac{a^{2}q^{1+|n|}}{bcde} )^{|k|} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/de; q)_{n_{i}}}{(ax_{i}q/d; q)_{n_{i}} (ax_{i}q/e; q)_{n_{i}}} ]

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{(de/ax_{i}; q)_{|k|−k_{i}} (aq/bcx_{i}; q)_{|k|}}{(aq/bcx_{i}; q)_{|k|−k_{i}} (deq^{−n_{i}}/ax_{i}; q)_{|k|}} ]

× \frac{(d; q)_{|k|} (e; q)_{|k|}}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ). (VI.4)

Proof. We first replace λ by qa^{2}/bcd in Theorem V.15, and let d → ∞. Finally, relabel f ↦ d in the resulting identity to obtain (VI.4).

Remark VI.5. Theorem VI.3 concerns series summed over an r-rectangle. By using a polynomial argument similar to the proof of Theorem V.20, we may find another A_{r} extension of Theorem VI.1, where the series involved are summed over an r-tetrahedron. This may also be accomplished by taking the limit as d → ∞ in (V.21).

Remark VI.6. Theorem VI.3 is different from any of the generalizations of Theorem VI.1 which appears in [56]. The series on the left hand side of (VI.4) is the same as two such transformations appearing in [59, Theorems A10 and 5.1]. By comparing the other two sides we obtain generalizations of the Sears _{4}ϕ_{3} transformations [69], [25, equations (III.15) and (III.16)], see [59] for an example of similar calculations.

The rest of our extensions of Theorem VI.1 concern D_{r} series. After taking limits, we sometimes use

\binom{|k|}{2} − e_{2}(k) = \binom{k_{1}}{2} + \binom{k_{2}}{2} + ⋯ + \binom{k_{r}}{2}

to simplify the powers of q.

Theorem VI.7 (A D_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.8) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} (ax_{i}x_{j}q/c; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/c; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (c/x_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (c/x_{i}; q)_{|k|−k_{i}}} \frac{∏_{i=1}^{r}(bx_{i}; q)_{k_{i}}}{(aq/b; q)_{|k|}}

× \frac{(d; q)_{|k|} (e; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/d; q)_{k_{i}} (ax_{i}q/e; q)_{k_{i}}]} (\frac{a^{2}q^{1+|n|}}{bcde} )^{|k|} q^{∑_{i=1}^{r} i k_{i}}q^{−2e_{2}(k)} ∏_{i=1}^{r} x_{i}^{2k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/de; q)_{n_{i}}}{(ax_{i}q/d; q)_{n_{i}} (ax_{i}q/e; q)_{n_{i}}} ]

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(ax_{i}x_{j}q/c; q)_{k_{i}+k_{j}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/c; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(de/ax_{i}; q)_{|k|−k_{i}}}{(deq^{−n_{i}}/ax_{i}; q)_{|k|}}

× ∏_{i=1}^{r}(ax_{i}q/bc; q)_{k_{i}} · \frac{(d; q)_{|k|} (e; q)_{|k|}}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (VI.8)

where e_{2}(k) is the second elementary symmetric function of k.

Proof. We first replace λ by qa^{2}/bcd in Theorem V.15, and let c → ∞. Finally, relabel d ↦ c, and f ↦ d in the resulting identity to obtain (VI.8).

Remark VI.9. Theorem VI.7 concerns series summed over an r-rectangle. By using a polynomial argument similar to the proof of Theorem V.20, we may find another transformation formula between series summed over an r-tetrahedron. This may also be accomplished by taking the limit as c → ∞ in (V.21).

Next, we consider the special cases of the equivalent formulation of Theorem V.15, obtained by replacing e by λaq/ef in (V.16).

Theorem VI.10 (A D_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.11) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} \frac{(ax_{i}x_{j}q/c; q)_{k_{i}+k_{j}}}{(ex_{i}x_{j}; q)_{k_{i}+k_{j}}} ∏_{i=1}^{r} \frac{(aq/ex_{i}; q)_{|k|−k_{i}}}{(c/x_{i}; q)_{|k|−k_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ex_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/c; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (c/x_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (aq^{1−n_{i}}/ex_{i}; q)_{|k|}}

× ∏_{i=1}^{r} [\frac{(bx_{i}; q)_{k_{i}}}{(ax_{i}q/d; q)_{k_{i}}} ] · \frac{(d; q)_{|k|}}{(aq/b; q)_{|k|}} (\frac{qa^{2}}{bcde})^{|k|} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}} (dex_{i}/a; q)_{n_{i}}}{(ex_{i}/a; q)_{n_{i}} (ax_{i}q/d; q)_{n_{i}}} ] (\frac{1}{d})^{|n|}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r} \frac{(ax_{i}x_{j}q/c; q)_{k_{i}+k_{j}}}{(ex_{i}x_{j}; q)_{k_{i}+k_{j}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ex_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/c; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{(ax_{i}q/bc; q)_{k_{i}}}{(dex_{i}/a; q)_{k_{i}}} ] · \frac{(d; q)_{|k|}}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ). (VI.11)

Proof. We first replace e by λaq/ef in (V.16) to obtain an equivalent formulation of Theorem V.15. Next, we replace λ by qa^{2}/bcd, and let c → ∞. Finally, relabel f ↦ d and d ↦ c in the resulting identity to obtain (VI.11).

Remark VI.12. Theorem VI.10 concerns series summed over an r-rectangle. By using a polynomial argument similar to the proof of Theorem V.20, we may find another transformation formula between series summed over an r-tetrahedron. This may also be accomplished by first replacing e by λaq/ef_{1} ⋯ f_{r} in (V.21), and then taking the limit as c → ∞ in (V.21).

Theorem VI.13 (A D_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.14) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} \frac{1}{(ex_{i}x_{j}; q)_{k_{i}+k_{j}}} ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ex_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/ex_{i}; q)_{|k|−k_{i}}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (aq^{1−n_{i}}/ex_{i}; q)_{|k|}} \frac{∏_{i=1}^{r} [(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}}

× \frac{(d; q)_{|k|}}{∏_{i=1}^{r}(ax_{i}q/d; q)_{k_{i}}} (\frac{qa^{2}}{bcde})^{|k|} q^{∑_{i=1}^{r} i k_{i}}q^{2e_{2}(k)} ∏_{i=1}^{r} x_{i}^{−2k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}} (dex_{i}/a; q)_{n_{i}}}{(ex_{i}/a; q)_{n_{i}} (ax_{i}q/d; q)_{n_{i}}} ] (\frac{1}{d})^{|n|}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r} \frac{1}{(ex_{i}x_{j}; q)_{k_{i}+k_{j}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ex_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(aq/bcx_{i}; q)_{|k|}}{(aq/bcx_{i}; q)_{|k|−k_{i}}}

× ∏_{i=1}^{r} [\frac{1}{(dex_{i}/a; q)_{k_{i}}} ] · \frac{(d; q)_{|k|}}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (VI.14)

where e_{2}(k) is the second elementary symmetric function of k.

Proof. We first replace e by λaq/ef in (V.16) to obtain an equivalent formulation of Theorem V.15. Next, we replace λ by qa^{2}/bcd, and let d → ∞. Finally, relabel f ↦ d in the resulting identity to obtain (VI.14).

Remark VI.15. Theorem VI.13 concerns series summed over an r-rectangle. By using a polynomial argument similar to the proof of Theorem V.20, we may find another transformation formula between series summed over an r-tetrahedron. This may also be accomplished by first replacing e by λaq/ef_{1} ⋯ f_{r} in (V.21), and then taking the limit as d → ∞ in the resulting identity.

Next, we obtain the Watson’s transformations which follow from Theorem V.22. If we take the limit as d → ∞ in Theorem V.22, we obtain one of Milne’s transformations, Theorem VI.25, see [61, Theorem 4.5].

Another limiting case yields:

Theorem VI.16 (A D_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.17) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} (ax_{i}x_{j}q/c; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/c; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (c/x_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (c/x_{i}; q)_{|k|−k_{i}}} \frac{∏_{i=1}^{r} [(bx_{i}; q)_{k_{i}} (ex_{i}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/e; q)_{|k|}}

× \frac{(d; q)_{|k|}}{∏_{i=1}^{r}(ax_{i}q/d)_{k_{i}}} (\frac{a^{2}q^{1+|n|}}{bcde} )^{|k|} q^{∑_{i=1}^{r} i k_{i}}q^{−e_{2}(k)} ∏_{i=1}^{r} x_{i}^{k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}}}{(ax_{i}q/d; q)_{n_{i}}} ] \frac{(aq/de; q)_{|n|}}{(aq/e; q)_{|n|}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(ax_{i}x_{j}q/c; q)_{k_{i}+k_{j}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/c; q)_{k_{i}}} ∏_{i=1}^{r} [(ax_{i}q/bc; q)_{k_{i}} (ex_{i}; q)_{k_{i}}]

× \frac{(d; q)_{|k|}}{(aq/b; q)_{|k|} (deq^{−|n|}/a; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (VI.17)

where e_{2}(k) is the second elementary symmetric function of k.

Proof. We first replace λ by qa^{2}/bcd in Theorem V.22, and let c → ∞. Finally, relabel d ↦ c, and f ↦ d in the resulting identity to obtain (VI.17).

Remark VI.18. Theorem VI.16 concerns series summed over an r-rectangle. By using a polynomial argument similar to the proof of Theorem V.24, we may find another transformation formula between series summed over an r-tetrahedron. This may also be accomplished by taking the limit as c → ∞ in (V.25).

Next, we consider the special cases of the equivalent formulation of Theorem V.22, obtained by replacing f by λaq^{1+|n|}/ef in (V.23). If we take the limit as c → ∞ in the resulting identity, we obtain a D_{r} generalization of Theorem VI.1 found by Bhatnagar [12], see Theorem VI.35.

Instead, if we take the limit as d → ∞, we obtain an A_{r} transformation theorem found by Milne [56], see Theorem VI.28. Further, if we take the limit as d → ∞ in Theorem V.26, we obtain an A_{r} generalization of Theorem VI.1 found by Milne [56], see Theorem VI.30. Leininger and Milne [46] have found some elegant applications of this case of Theorem V.26. The b → ∞ case of Theorem V.26 was also found by Milne [56], see Theorem VI.33.

Finally, we consider the equivalent formulation of Theorem V.26, obtained by replacing e by λaq/ef in (V.27).

Theorem VI.19 (A D_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.20) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} \frac{1}{(ex_{i}x_{j}; q)_{k_{i}+k_{j}}} ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ex_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/ex_{i}; q)_{|k|−k_{i}}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (aq^{1−n_{i}}/ex_{i}; q)_{|k|}}

× \frac{(b; q)_{|k|} (c; q)_{|k|} (d; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}}(ax_{i}q/d; q)_{k_{i}}]} (\frac{qa^{2}}{bcde})^{|k|} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}} (dex_{i}/a; q)_{n_{i}}}{(ex_{i}/a; q)_{n_{i}} (ax_{i}q/d; q)_{n_{i}}} ] (\frac{1}{d})^{|n|}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r} \frac{1}{(ex_{i}x_{j}; q)_{k_{i}+k_{j}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ex_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{(ax_{i}q/bc; q)_{k_{i}}}{(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (dex_{i}/a; q)_{k_{i}}} ] · (d; q)_{|k|} q^{∑_{i=1}^{r} i k_{i}} ). (VI.20)

Proof. We first replace e by λaq/ef in (V.27) to obtain an equivalent formulation of Theorem V.26. Next, we replace λ by qa^{2}/bcd, and let b → ∞. Finally, relabel d ↦ b, f ↦ d in the resulting identity to obtain (VI.20).

Remark VI.21. Theorem VI.19 concerns series summed over an r-rectangle. By using a polynomial argument similar to the proof of Theorem V.28, we may find another transformation formula between series summed over an r-tetrahedron.

This may also be accomplished by first replacing e by λaq/ef_{1} ⋯ f_{r} in (V.29), and then taking the limit as b → ∞ in the resulting identity.

Theorem VI.22 (A D_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.23) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} \frac{1}{(ex_{i}x_{j}; q)_{k_{i}+k_{j}}} ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ex_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (aq/ex_{i}; q)_{|k|−k_{i}}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (aq^{1−n_{i}}/ex_{i}; q)_{|k|}} \frac{∏_{i=1}^{r}(bx_{i}; q)_{k_{i}}}{(aq/b; q)_{|k|}}

× \frac{(c; q)_{|k|} (d; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}}]} · (\frac{qa^{2}}{bcde})^{|k|} q^{∑_{i=1}^{r} i k_{i}}q^{e_{2}(k)} ∏_{i=1}^{r} x_{i}^{−k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}} (dex_{i}/a; q)_{n_{i}}}{(ex_{i}/a; q)_{n_{i}} (ax_{i}q/d; q)_{n_{i}}} ] (\frac{1}{d})^{|n|}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r} \frac{1}{(ex_{i}x_{j}; q)_{k_{i}+k_{j}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ex_{i}x_{j}q^{n_{j}}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{1}{(ax_{i}q/c; q)_{k_{i}} (dex_{i}/a; q)_{k_{i}}} ] \frac{(aq/bc; q)_{|k|} (d; q)_{|k|}}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (VI.23)

where e_{2}(k) is the second elementary symmetric function of k.

Proof. We first replace e by λaq/ef in (V.27) to obtain an equivalent formulation of Theorem V.26. Next, we replace λ by qa^{2}/bcd, and let d → ∞. Finally, relabel f ↦ d in the resulting identity to obtain (VI.23).

Remark VI.24. Theorem VI.22 concerns series summed over an r-rectangle. By using a polynomial argument similar to the proof of Theorem V.28, we may find another transformation formula between series summed over an r-tetrahedron.

This may also be accomplished by first replacing e by λaq/ef_{1} ⋯ f_{r} in (V.29), and then taking the limit as d → ∞ in the resulting identity.

Before proceeding to some of Milne’s Watson’s transformations, we indicate the effect of reversing the sum on the various generalizations of Theorem VI.1 presented above. Theorem VI.3 and Theorem VI.10 are invariant if we reverse the series on both sides of the transformation, and relabel parameters. If we do the same for Theorem VI.7, we obtain Theorem VI.13, and vice-versa. Similarly, Theorem VI.16 and Theorem VI.22 are transformed into each other by reversing series. Finally, Theorem VI.19 and Bhatnagar’s [12] D_{r} Watson’s transformation, Theorem VI.35, are similarly seen to be equivalent.

Theorem VI.25 ((Milne) An A_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.26) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|}}

× \frac{∏_{i=1}^{r} [(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}} (ex_{i}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (aq/e; q)_{|k|}} \frac{(d; q)_{|k|}}{∏_{i=1}^{r}(ax_{i}q/d; q)_{k_{i}}}

× (\frac{a^{2}q^{1+|n|}}{bcde} )^{|k|} q^{∑_{i=1}^{r} i k_{i}} q^{e_{2}(k)} ∏_{i=1}^{r} x_{i}^{−k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}}}{(ax_{i}q/d; q)_{n_{i}}} ] \frac{(aq/de; q)_{|n|}}{(aq/e; q)_{|n|}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{(aq/bcx_{i}; q)_{|k|} (ex_{i}; q)_{k_{i}}}{(aq/bcx_{i}; q)_{|k|−k_{i}}} ]

× \frac{(d; q)_{|k|}}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (deq^{−|n|}/a; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (VI.26)

where e_{2}(k) is the second elementary symmetric function of k.

Proof. We first replace λ by qa^{2}/bcd in Theorem V.22, and let d → ∞. Finally, relabel f ↦ d in the resulting identity to obtain (VI.26).

Remark VI.27. The transformation formula in Theorem VI.25 is one of Milne’s A_{r} Watson’s transformations, see [61, Theorem 4.5].

The following three A_{r} transformation formulas have initially been derived by Milne, see [56], but they also follow immediately from our results.

Theorem VI.28 ((Milne) An A_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.29) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|}}

× \frac{∏_{i=1}^{r} [(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}} (dx_{i}; q)_{k_{i}} (ex_{i}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (aq/d; q)_{|k|} (aq/e; q)_{|k|}}

× (\frac{a^{2}q^{1+|n|}}{bcde} )^{|k|} q^{∑_{i=1}^{r} i k_{i}} q^{2e_{2}(k)} ∏_{i=1}^{r} x_{i}^{−2k_{i}} )

= \frac{∏_{i=1}^{r} [(ax_{i}q; q)_{n_{i}} (aq^{1+|n|−n_{i}}/dex_{i}; q)_{n_{i}}]}{(aq/d; q)_{|n|} (aq/e; q)_{|n|}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{(aq/bcx_{i}; q)_{|k|}}{(aq/bcx_{i}; q)_{|k|−k_{i}} (dex_{i}q^{−|n|}/a; q)_{k_{i}}} ]

× \frac{∏_{i=1}^{r} [(dx_{i}; q)_{k_{i}} (ex_{i}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (VI.29)

where e_{2}(k) is the second elementary symmetric function of k.

Proof. We first replace f by λaq^{1+|n|}/ef in (V.23) to obtain an equivalent formulation of Theorem V.22. Next, we replace λ by qa^{2}/bcd, and let d → ∞. Finally, relabel f ↦ d in the resulting identity to obtain (VI.29).

Theorem VI.30 ((Milne) An A_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.31) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|}}

× \frac{∏_{i=1}^{r}(bx_{i}; q)_{k_{i}}}{(aq/b; q)_{|k|}} \frac{(c; q)_{|k|} (d; q)_{|k|} (e; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}} (ax_{i}q/e; q)_{k_{i}}]}

× (\frac{a^{2}q^{1+|n|}}{bcde} )^{|k|} q^{∑_{i=1}^{r} i k_{i}} q^{−e_{2}(k)} ∏_{i=1}^{r} x_{i}^{k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/de; q)_{n_{i}}}{(ax_{i}q/d; q)_{n_{i}} (ax_{i}q/e; q)_{n_{i}}} ]

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{(de/ax_{i}; q)_{|k|−k_{i}}}{(deq^{−n_{i}}/ax_{i}; q)_{|k|} (ax_{i}q/c; q)_{k_{i}}} ]

× \frac{(aq/bc; q)_{|k|} (d; q)_{|k|} (e; q)_{|k|}}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ), (VI.31)

where e_{2}(k) is the second elementary symmetric function of k.

Proof. We first replace λ by qa^{2}/bcd in Theorem V.26, and let d → ∞. Finally, relabel f ↦ d in the resulting identity to obtain (VI.31).

Remark VI.32. Leininger and Milne [46] have found some elegant applications of Theorem VI.30.

Theorem VI.33 ((Milne) An A_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.34) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|}}

× \frac{(b; q)_{|k|} (c; q)_{|k|} (d; q)_{|k|} (e; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}} (ax_{i}q/e; q)_{k_{i}}]}

× (\frac{a^{2}q^{1+|n|}}{bcde} )^{|k|} q^{∑_{i=1}^{r} i k_{i}} q^{−2e_{2}(k)} ∏_{i=1}^{r} x_{i}^{2k_{i}} )

= ∏_{i=1}^{r} [\frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/de; q)_{n_{i}}}{(ax_{i}q/d; q)_{n_{i}} (ax_{i}q/e; q)_{n_{i}}} ]

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{(de/ax_{i}; q)_{|k|−k_{i}} (ax_{i}q/bc; q)_{k_{i}}}{(deq^{−n_{i}}/ax_{i}; q)_{|k|}} ]

× \frac{(d; q)_{|k|} (e; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}}]} q^{∑_{i=1}^{r} i k_{i}} ), (VI.34)

where e_{2}(k) is the second elementary symmetric function of k.

Proof. We first replace λ by qa^{2}/bcd in Theorem V.26, and let b → ∞. Finally, relabel d ↦ b and f ↦ d in the resulting identity to obtain (VI.34).

Finally, we state a D_{r} transformation formula which was recently found by Bhatnagar [12].

Theorem VI.35 ((Bhatnagar) A D_{r} Watson’s transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.36) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r} (ax_{i}x_{j}q/c; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/c; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (c/x_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (c/x_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(bx_{i}; q)_{k_{i}} (dx_{i}; q)_{k_{i}} (ex_{i}; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/d; q)_{|k|} (aq/e; q)_{|k|}} (\frac{a^{2}q^{1+|n|}}{bcde} )^{|k|} q^{∑_{i=1}^{r} i k_{i}} )

= \frac{∏_{i=1}^{r} [(ax_{i}q; q)_{n_{i}} (aq^{1+|n|−n_{i}}/dex_{i}; q)_{n_{i}}]}{(aq/d; q)_{|n|} (aq/e; q)_{|n|}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(ax_{i}x_{j}q/c; q)_{k_{i}+k_{j}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q/c; q)_{k_{i}}}

× ∏_{i=1}^{r} [\frac{(ax_{i}q/bc; q)_{k_{i}} (dx_{i}; q)_{k_{i}} (ex_{i}; q)_{k_{i}}}{(dex_{i}q^{−|n|}/a; q)_{k_{i}}} ] · \frac{1}{(aq/b; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ). (VI.36)

Proof. We first replace f by λaq^{1+|n|}/ef in (V.23) to obtain an equivalent formulation of Theorem V.22. Next, we replace λ by qa^{2}/bcd, and let c → ∞. Finally, relabel f ↦ d and d ↦ c in the resulting identity to obtain (VI.36).

In the following, we indicate how multivariable Sears’ [69] transformations can be worked out. Many of these follow from our Watson’s transformations of this chapter, respectively from our Bailey’s _{10}ϕ_{9} transformations of chapter V. Special cases of our generalizations of Watson’s _{8}ϕ_{7} transformation include A_{r}, C_{r}, and D_{r} generalizations of Sears’ transformation found by Milne and Newcomb [59], [61], Milne and Lilly [58], and Bhatnagar [12], respectively. In addition to recovering all such transformations known previously, we can also use our results to obtain several new A_{r} and D_{r} generalizations of Sears’ theorem.

Sears’ formula follows from Watson’s _{8}ϕ_{7} transformation in one of many ways. Observe that the _{8}ϕ_{7} on the left side of (VI.2) is termwise unchanged if we interchange c with d, b with e. Equate the resulting sums corresponding to the right side of (VI.2), and then relabel the parameters a ↦ bcq^{−n}/d, d ↦ e/a, and e ↦ bcq^{1−n}/de. After some simplification we obtain [25, equation (2.10.4), explicitly]:

Theorem VI.37 (Sears’ classical _{4}ϕ_{3} transformation). Let a, b, c, d, and e be indeterminate, let n be a nonnegative integer, and suppose that none of the denominators in (VI.38) vanish. Then

4ϕ_{3} [\begin{matrix}q^{−n}, a, b, c \\ d, e, abcq^{1−n}/de \end{matrix}; q, q] = \frac{(e/a; q)_{n} (de/bc; q)_{n}}{(e; q)_{n} (de/abc; q)_{n}} _{4}ϕ_{3} [\begin{matrix}q^{−n}, a, d/b, d/c \\ d, aq^{1−n}/e, de/bc \end{matrix}; q, q] .

(VI.38)

This is not the only the only way to obtain Theorem VI.37 from (VI.2). We may also interchange other parameters, two by two, first, equate the_{4}ϕ_{3} sides as before, and then relabel parameters. In the classical case, due to symmetry all these cases lead to the same identity, up to a trivial relabelling of parameters. However, in our multivariable generalizations of Watson’s formula, some of its symmetry is broken, and we obtain many different generalizations of Sears’ transformation. Note that because of the broken symmetry in our _{10}ϕ_{9} transformations, we had also obtained, as limiting cases, different multivariable generalizations of Watson’s transformation formula.

But it is also possible to obtain Sears’_{4}ϕ_{3} transformation, (VI.38), from Bailey’s 10ϕ_{9} transformation, (V.3), directly. Replace b by aq/b and e by aq/e in (V.3), and then take the limit a → 0. After relabelling of parameters, b ↦ d, d ↦ b, and f ↦ a, we obtain (VI.38). Again, different limits lead to Sears’ transformation, where distinct identities might be obtained in the multidimensional cases.

A complete list of all possible Sears’ transformations would be even longer than a list including all Watson’s transformations. For illustration, we only give two multidimensional Sears’ transformations, both of which follow from Theorem V.15.

Theorem VI.39 (An A_{r} Sears’ transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.40) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(abcq/dex_{i}; q)_{|k|−k_{i}}}{(b/x_{i}; q)_{|k|−k_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(b/x_{i}; q)_{|k|}}{(abcq^{1−n_{i}}/dex_{i}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(cx_{i}; q)_{k_{i}}}{(ex_{i}; q)_{k_{i}}} · \frac{(a; q)_{|k|}}{(d; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ex_{i}/a; q)_{n_{i}} (dex_{i}/bc; q)_{n_{i}}}{(ex_{i}; q)_{n_{i}} (dex_{i}q/abc; q)_{n_{i}}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(aq/ex_{i}; q)_{|k|−k_{i}}}{(d/cx_{i}; q)_{|k|−k_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(d/cx_{i}; q)_{|k|}}{(aq^{1−n_{i}}/ex_{i}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(dx_{i}/b; q)_{k_{i}}}{(dex_{i}/bc; q)_{k_{i}}} · \frac{(a; q)_{|k|}}{(d; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} ). (VI.40)

Proof. We first replace b by aq/b and e by aq/e in (V.16), and then take the limit a → 0. After relabelling of parameters, b ↦ d, d ↦ b, and f ↦ a, we obtain (VI.40).

Theorem VI.41 (A D_{r} Sears’ transformation). Let x_{1}, …, x_{r}, a, b, c, d, and e be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (VI.42) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(dx_{i}x_{j}; q)_{k_{i}+k_{j}}

× ∏_{i=1}^{r} \frac{(abcq/dex_{i}; q)_{|k|−k_{i}}}{(abcq^{1−n_{i}}/dex_{i}; q)_{|k|}} ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (dx_{i}x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}}}{(ex_{i}; q)_{k_{i}}} · (a; q)_{|k|} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ex_{i}/a; q)_{n_{i}} (dex_{i}/bc; q)_{n_{i}}}{(ex_{i}; q)_{n_{i}} (dex_{i}q/abc; q)_{n_{i}}}

×_{∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r}( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{1≤i<j≤r}(dx_{i}x_{j}; q)_{k_{i}+k_{j}}

× ∏_{i=1}^{r} \frac{(aq/ex_{i}; q)_{|k|−k_{i}}}{(aq^{1−n_{i}}/ex_{i}; q)_{|k|}} ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (dx_{i}x_{j}; q)_{k_{i}}}

× ∏_{i=1}^{r} \frac{(dx_{i}/b; q)_{k_{i}} (dx_{i}/c; q)_{k_{i}}}{(dex_{i}/bc; q)_{k_{i}}} · (a; q)_{|k|} q^{∑_{i=1}^{r} i k_{i}} ). (VI.42)

Proof. We first replace d by aq/d and e by aq/e in (V.16), and then take the limit a → 0. After relabelling, f ↦ a, we obtain (VI.42).

CHAPTER VII More multidimensional summation theorems

In this chapter, which can be read independently from the previous ones, we derive multiple generalizations of many of the classical basic hypergeometric summation formulas. These extensions of a new, natural, kind of multidimensional basic hypergeometric series are also associated to root systems of classical type. We proceed by combining the classical (one-dimensional) summation formulas with certain determinant evaluations. Our theorems include A_{r} extensions of Ramanujan’s bilateral _{1}ψ_{1} sum, C_{r} extensions of Bailey’s very-well-poised _{6}ψ_{6} summation, and a C_{r} extension of Jackson’s very-well-poised balanced _{8}ϕ_{7} summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A_{r}, B_{r}, C_{r}, and D_{r}, respectively, of Chu’s bilateral transformation formula for basic hypergeometric series of Gasper-Karlsson-Minton type. Limiting cases of our series identities include multidimensional generalizations of many of the most important summation theorems of the classical theory of basic hypergeometric series. As explicit examples we provide C_{r} terminating and nonterminating 6ϕ_{5} summation formulas, and an A_{r} extension of the q-Pfaff-Saalschütz summation. This chapter is part of a forthcoming paper [68].

Recently, Gustafson and Krattenthaler [32, Theorem 1.15] discovered an A_{r} (or equivalently U(r + 1)) extension of Ramanujan’s bilateral _{1}ψ_{1} sum. The work in their paper involved A_{r} series of a new kind. In particular, Gustafson and Krattenthaler’s A_{r} _{1}ψ_{1} sum is different from Milne’s [52].

Before we review Ramanujan’s _{1}ψ_{1} sum and Gustafson and Krattenthaler’s extension thereof we introduce the Gasper-Rahman notation

(a_{1}, …, a_{m}; q)_{k} ≡ (a_{1}; q)_{k} (a_{2}; q)_{k} ⋯ (a_{m}; q)_{k},

where the rising q-factorial (a; q)_{k} is defined in (IV.3), for simplifying our displays in this chapter.

Ramanujan’s classical _{1}ψ_{1} summation formula (see [35] or [25, (5.2.1)]) reads

Theorem VII.1 (Ramanujan’s classical _{1}ψ_{1} sum). Let a, b, and z be indeterminate, and suppose that none of the denominators in (VII.2) vanish. Then

∑_{k=−∞}^{∞} \frac{(a; q)_{k}}{(b; q)_{k}} z^{k} = \frac{(q, b/a, az, q/az; q)_{∞}}{(b, q/a, z, b/az; q)_{∞}} ,  (VII.2)

provided the series terminates or |q| < 1 and |b/a| < |z| < 1.

Remark VII.3. Ramanujan’s bilateral _{1}ψ_{1} summation formula is one of the most fundamental formulas of the theory of basic hypergeometric series. A simple and elegant proof was given by Ismail [38] who noted that Theorem VII.1 is an immediate consequence of the q-binomial theorem (which is the b = q case of Theorem VII.1), and analytic continuation.

Gustafson and Krattenthaler’s A_{r} _{1}ψ_{1} summation formula [32, Theorem 1.15] can be stated as follows. Here and in the following we use the notation |k| = k_{1} + k_{2} + ⋯ + k_{r}.

Theorem VII.4 (Gustafson and Krattenthaler). Let x_{1}, …, x_{r}, a, b, and z be indeterminate, let r ≥ 1, and suppose that none of the denominators in (VII.5) vanish. Then

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(a; q)_{k_{i}}}{(b; q)_{k_{i}}} · z^{|k|} q^{∑_{i=1}^{r}(i−r) k_{i}} )

= ∏_{i=1}^{r} \frac{(q, b/a, azq^{1−i}, q^{i}/az; q)_{∞}}{(b, q/a, zq^{1−i}, bq^{i−1}/az; q)_{∞}} , (VII.5)

provided the series terminates or |b/a| < |z| < |q|^{r−1} < 1.

Gustafson and Krattenthaler proved Theorem VII.4 by a combination of the classical r = 1 case (VII.2) and the Vandermonde determinant evaluation. One of the main purposes of this chapter is to provide some more A_{r} extensions of Ramanujan’s _{1}ψ_{1} summation formula. But we are also able to derive C_{r} extensions of Bailey’s bilateral very-well-poised _{6}ψ_{6} summation, and Jackson’s balanced very-well-poised _{8}ϕ_{7} summation formula, respectively. Furthermore, at the end of this chapter, we combine determinant evaluations with Chu’s [18] remarkable bilateral transformation formula of Gasper-Karlsson-Minton type [23], [42], [60], [25, sec. 1.9] to deduce multiple versions of Chu’s identity. It is suprising that we obtain identities associated to various root systems of classical type, and we may also employ different bases, q_{1}, …, q_{r}, in our series (see Theorem VII.46). It is also possible to obtain other multiple extensions of Chu’s transformation formula by using other determinants in our derivation.

Proceeding by essentially the same method as Gustafson and Krattenthaler in the proof of their A_{r} _{1}ψ_{1} summation theorem our derivations in this chapter require certain determinant evaluations which are more general than the classical Vandermonde determinant evaluation. One of the determinant evaluations we utilize, Lemma A.22, comes from a determinant lemma [44, Lemma 34] (see Lemma A.24 in Appendix A) which has been successfully used by Krattenthaler in the computation of generating functions for plane partitions and tableaux. This determinant lemma has also been used by Gessel and Krattenthaler [28] for deriving several A_{r} basic hypergeometric series identities. Independently, a special case of Lemma A.22 was involved in [77] in the computation of biorthogonal rational functions.

The particular technique in this chapter has already been used in [33] to provide new proofs and generalizations of the A_{r} extensions of Heine’s _{2}ϕ_{1} transformations which have been discovered in [32]. In fact, the reading of [33] was the starting point of the author’s investigations for identities of this kind of series. We believe that the method of this chapter, which is entirely elementary, will be useful for proving other multidimensional series identities as well [68].

Our A_{r} extensions of Ramanujan’s _{1}ψ_{1} summation formula, (VII.2), are the following.

Theorem VII.6 (A_{r} _{1}ψ_{1} summations). Let x_{1}, …, x_{r}, a_{1}, …, a_{r}, b_{1} …, b_{r}, z_{1}, …, z_{r}, a, b, and z be indeterminate, let r ≥ 1, and suppose that none of the denominators in (VII.7), (VII.8), or (VII.9) vanish. Then

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(ax_{i}; q)_{k_{i}}}{(b_{i}; q)_{k_{i}}} · z^{|k|} q^{∑_{i=1}^{r}(i−r) k_{i}} )

= q^{−\binom{r}{2}} ∏_{1≤i<j≤r} (\frac{b_{i}/x_{i} − b_{j}/x_{j}}{1/x_{i} − 1/x_{j}} ) ∏_{i=1}^{r} \frac{(q, b_{i}/ax_{i}, ax_{i}z, q/ax_{i}z; q)_{∞}}{(b_{i}, q/ax_{i}, zq^{1−i}, b_{i}/ax_{i}z; q)_{∞}} , (VII.7)

provided the series terminates or |b_{i}/ax_{i}| < |z| < |q|^{r−1} < 1 for i = 1, …, r,

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(a_{i}; q)_{k_{i}}}{(bx_{i}; q)_{k_{i}}} · z^{|k|} q^{∑_{i=1}^{r}(i−r) k_{i}} )

= ∏_{1≤i<j≤r} (\frac{1 − a_{j}x_{i}/a_{i}x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(q, bx_{i}/a_{i}, a_{i}zq^{1−i}, q^{i}/a_{i}z; q)_{∞}}{(bx_{i}, q/a_{i}, zq^{1−i}, bx_{i}/a_{i}z; q)_{∞}} , (VII.8)

provided the series terminates or |bx_{i}/a_{i}| < |z| < |q|^{r−1} < 1 for i = 1, …, r,

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(ax_{i}; q)_{k_{i}}}{(bx_{i}; q)_{k_{i}}} z_{i}^{k_{i}} · q^{∑_{i=1}^{r}(i−r) k_{i}} )

= ∏_{1≤i<j≤r} (\frac{1 − z_{j}/z_{i}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(q, bq^{1−i}/a, ax_{i}z_{i}q^{1−i}, q^{i}/ax_{i}z_{i}; q)_{∞}}{(bx_{i}, q/ax_{i}, z_{i}q^{1−r}, b/az_{i}; q)_{∞}} , (VII.9)

provided the series terminates or |b/a| < |z_{i}| < |q|^{r−1} < 1 for i = 1, …, r.

Proof. We start with the sum

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(a_{i}; q)_{k_{i}}}{(b_{i}; q)_{k_{i}}} z_{i}^{k_{i}} · q^{∑_{i=1}^{r}(i−r) k_{i}} ),(VII.10)

and specialize the parameters a_{i}, b_{i}, z_{i} later. We have

q^{∑_{i=1}^{r}(i−r) k_{i}} ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) = ∏_{1≤i<j≤r} (\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} )

= ∏_{1≤i<j≤r}(1/x_{i} − 1/x_{j})^{−1} det_{1≤s,t≤r} ((q^{−k_{s}}/x_{s})^{r−t}) ,

the last equation due to the Vandermonde determinant evaluation. Hence we may write (VII.10), when multiplied by ∏_{1≤i<j≤r}(1/x_{i} − 1/x_{j}), as

det_{1≤s,t≤r} (x_{s}^{t−r} ∑_{k_{s}=−∞}^{∞} \frac{(a_{s}; q)_{k_{s}}}{(b_{s}; q)_{k_{s}}} (z_{s}q^{t−r})^{k_{s}} ) .

Now, to the sum inside the determinant we apply Ramanujan’s classical _{1}ψ_{1} summation, (VII.2), with a ↦ a_{s}, b ↦ b_{s}, and z ↦ z_{s}q^{t−r}. Thus we obtain

det_{1≤s,t≤r} (x_{s}^{t−r} \frac{(q, b_{s}/a_{s}, a_{s}z_{s}q^{t−r}, q^{1+r−t}/a_{s}z_{s}; q)_{∞}}{(b_{s}, q/a_{s}, z_{s}q^{t−r}, b_{s}q^{r−t}/a_{s}z_{s}; q)_{∞}} ) .

Now, by using linearity of the determinant with respect to rows, we take some factors out of the determinant and obtain

∏_{i=1}^{r} \frac{(q, b_{i}/a_{i}, a_{i}z_{i}, q/a_{i}z_{i}; q)_{∞}}{(b_{i}, q/a_{i}, z_{i}, b_{i}/a_{i}z_{i}; q)_{∞}} det_{1≤s,t≤r} ((\frac{x_{s}}{a_{s}} )^{t−r} \frac{(b_{s}/a_{s}z_{s}; q)_{r−t}}{(q/z_{s}; q)_{r−t}} ) . (VII.11)

The determinant in (VII.11) cannot be evaluated in closed form in general. But we can choose different specializations of the parameters a_{s}, b_{s}, and z_{s}, for s = 1, …, r, for which the determinant can be reduced to a product by means of Lemma A.22.

The simplest choice is a_{s} ≡ a, b_{s} ≡ b, and z_{s} ≡ z. In this case the determinant in (VII.11) equals

a^{\binom{r}{2}} ∏_{i=1}^{r} \frac{(b/az; q)_{r−i}}{(q/z; q)_{r−i}} det_{1≤s,t≤r} (x_{s}^{t−r}) ,

and the last determinant can be reduced to ∏_{1≤i<j≤r}(1/x_{i} − 1/x_{j}). Substituting these calculations and performing some other elementary manipulations, we easily recover (VII.5).

By choosing different specializations of the parameters in (VII.11) we will now prove the cases (VII.7), (VII.8), and (VII.9) of our theorem.

To prove (VII.7), we set a_{s} = ax_{s} and z_{s} ≡ z. In this case the determinant in (VII.11) equals

a^{\binom{r}{2}} ∏_{i=1}^{r}(q/z; q)_{r−i}^{−1} det_{1≤s,t≤r} ((b_{s}/ax_{s}z; q)_{r−t}) .

The determinant can be evaluated by means of Lemma A.22 with X_{s} ↦ b_{s}/x_{s}, A ↦ 1/az, B ↦ 0, and C ↦ 0. Subsequently, substituting our calculations and performing some other elementary manipulations leads to (VII.7).

To prove (VII.8), we set b_{s} = bx_{s} and z_{s} ≡ z. In this case the determinant in (VII.11) equals

∏_{i=1}^{r}(q/z; q)_{r−i}^{−1} det_{1≤s,t≤r} ((\frac{x_{s}}{a_{s}} )^{t−r} (bx_{s}/a_{s}z; q)_{r−t}) .  (VII.12)

The determinant can be evaluated by means of a limiting case of Lemma A.22. Namely, first multiply both sides of the C ↦ 0 case of (A.23) with (−B)^{\binom{r}{2}}q^{\binom{r}{3}} and then let B → ∞ in the resulting identity to see that

det_{1≤s,t≤r} (X_{s}^{t−r}(AX_{s}; q)_{r−t}) = ∏_{1≤i<j≤r}(1/X_{i} − 1/X_{j}).  (VII.13)

Now evaluate the determinant in (VII.12) by the X_{s} ↦ x_{s}/a_{s} and A ↦ b/z case of (VII.13). Substituting our calculations and performing some elementary manipulations, we easily deduce (VII.8).

To prove (VII.9), we set a_{s} = ax_{s} and b_{s} ≡ bx_{s}. In this case the determinant in (VII.11) equals

a^{\binom{r}{2}} det_{1≤s,t≤r} (\frac{(b/az_{s}; q)_{r−t}}{(q/z_{s}; q)_{r−t}} ) .

The determinant can be evaluated by means of Lemma A.22 with X_{s} ↦ 1/z_{s}, A ↦ b/a, B ↦ q, and C ↦ 0. Finally, substituting our calculations and performing some elementary manipulations, we arrive at (VII.9).

Remark VII.14. There are two other cases where the determinant in (VII.11) can be evaluated in closed form.

We may set b_{s} = bx_{s} and z_{s} ≡ zx_{s}/a_{s}. In this case the determinant in (VII.11) equals

∏_{i=1}^{r}(b/z; q)_{r−i} det_{1≤s,t≤r} ((\frac{a_{s}}{x_{s}} )^{r−t} (qa_{s}/x_{s}z; q)_{r−t}^{−1}) .  (VII.15)

The determinant can be evaluated by means of a limiting case of Lemma A.22. Namely, first multiply both sides of the C ↦ 0 case of (A.23) with (−A)^{−\binom{r}{2}}q^{−\binom{r}{3}} and then let A → ∞ in the resulting identity to see that

det_{1≤s,t≤r} (X_{s}^{r−t}(BX_{s}; q)_{r−t}^{−1}) = ∏_{i=1}^{r}(BX_{i}; q)_{r−1}^{−1} ∏_{1≤i<j≤r}(X_{i} − X_{j}). (VII.16)

Now evaluate the determinant in (VII.15) by the X_{s} ↦ a_{s}/x_{s} and B ↦ q/z case of (VII.16). Substituting our calculations and performing some other elementary manipulations leads to the following identity:

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(a_{i}; q)_{k_{i}}}{(bx_{i}; q)_{k_{i}}} (\frac{zx_{i}}{a_{i}} )^{k_{i}} · q^{∑_{i=1}^{r}(i−r) k_{i}} )

= ∏_{1≤i<j≤r} (\frac{x_{i}/a_{i} − x_{j}/a_{j}}{x_{i} − x_{j}} ) ∏_{i=1}^{r} \frac{(q, bx_{i}/a_{i}, zx_{i}q^{1−r}, q^{r}/zx_{i}; q)_{∞}}{(bx_{i}, q/a_{i}, zx_{i}q^{1−r}/a_{i}, bq^{i−1}/z; q)_{∞}} , (VII.17)

provided the series terminates or |bx_{i}/a_{i}| < |z| < |q|^{r−1} < 1 for i = 1, …, r.

Actually, (VII.17) is equivalent to (VII.7), since (VII.17) can be obtained by doing the replacements k_{i} ↦ −k_{i}, x_{i} ↦ 1/x_{i}, b_{i} ↦ q/a_{i}, for i = 1, …, r, a ↦ q/b, and z ↦ q^{r−1}b/z in (VII.7), and some elementary manipulations.

On the other hand, if we set a_{s} = ax_{s} and z_{s} ≡ zb_{s}/x_{s} in (VII.11) the determinant equals

a^{\binom{r}{2}} ∏_{i=1}^{r}(1/az; q)_{r−i} det_{1≤s,t≤r} ((qx_{s}/b_{s}z; q)_{r−t}^{−1}) .  (VII.18)

The determinant can be evaluated by means of the A, C → 0 limiting case of Lemma A.22, reading

det_{1≤s,t≤r} ((BX_{s}; q)_{r−t}^{−1}) = B^{\binom{r}{2}}q^{2\binom{r}{3}} ∏_{i=1}^{r}(BX_{i}; q)_{r−1}^{−1} ∏_{1≤i<j≤r}(X_{i} − X_{j}).(VII.19)

Now evaluate the determinant in (VII.18) by the X_{s} ↦ x_{s}/b_{s} and B ↦ q/z case of (VII.19). Again, substituting our calculations and performing some other elementary manipulations we arrive at the following summation:

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(ax_{i}; q)_{k_{i}}}{(b_{i}; q)_{k_{i}}} (\frac{zb_{i}}{x_{i}} )^{k_{i}} · q^{∑_{i=1}^{r}(i−r) k_{i}} )

= ∏_{1≤i<j≤r} (\frac{1 − b_{j}x_{i}/b_{i}x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(q, b_{i}/ax_{i}, azb_{i}q^{1−i}, q^{i}/azb_{i}; q)_{∞}}{(b_{i}, q/ax_{i}, zb_{i}q^{1−r}/x_{i}, q^{i−1}/az; q)_{∞}} , (VII.20)

provided the series terminates or |b_{i}/ax_{i}| < |z| < |q|^{r−1} < 1 for i = 1, …, r.

As in the case above we have not obtained a new identity, since (VII.20) is equivalent to (VII.8), where (VII.20) can be obtained by doing the replacements k_{i} ↦ −k_{i}, x_{i} ↦ 1/x_{i}, a_{i} ↦ q/b_{i}, for i = 1, …, r, b ↦ q/a, and z ↦ q^{r−1}a/z in (VII.8), and some elementary manipulations.

Remark VII.21. The choice b_{i} ≡ qa_{i} in (VII.10) is a special case of the summation formula implied by Theorem VII.46, where we even could have started with different bases q_{i} in the series (VII.10). In this case most of the factors in (VII.11) cancel.

Before we state the C_{r} extensions of Bailey’s_{6}ψ_{6} summation formula that we are going to prove, it might be convenient to recall Bailey’s original _{6}ψ_{6} summation. This is (cf. [9], [25, (5.3.1)])

Theorem VII.22 (Bailey’s classical _{6}ψ_{6} sum). Let a, b, c, d, and e be indeterminate, and suppose that none of the denominators in (VII.23) vanish. Then

∑_{k=−∞}^{∞} \frac{1 − aq^{2k}}{1 − a} \frac{(b, c, d, e; q)_{k}}{(aq/b, aq/c, aq/d, aq/e; q)_{k}} (\frac{qa^{2}}{bcde})^{k}

= \frac{(aq, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de, q, q/a; q)_{∞}}{(aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e, qa^{2}/bcde; q)_{∞}} , (VII.23)

provided the series terminates or |q| < 1 and |qa^{2}/bcde| < 1.

Remark VII.24. Andrews [1] discusses some applications of Bailey’s very-well-poised _{6}ψ_{6} summation formula to number theory.

Our C_{r} extensions of Bailey’s bilateral very-well poised _{6}ψ_{6} summation formula are the following.

Theorem VII.25 (C_{r} _{6}ψ_{6} summations). Let x_{1}, …, x_{r}, e_{1}, …, e_{r}, a, b, c, d, and e be indeterminate, let r ≥ 1, and suppose that none of the denominators in (VII.26) or (VII.27) vanish. Then

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} )

88 VII. MORE MULTIDIMENSIONAL SUMMATION THEOREMS

× ∏_{i=1}^{r} \frac{(bx_{i}, cx_{i}, dx_{i}, e_{i}x_{i}; q)_{k_{i}}}{(ax_{i}q/b, ax_{i}q/c, ax_{i}q/d, ax_{i}q/e_{i}; q)_{k_{i}}} (\frac{a^{2}q}{bcde_{i}} )^{k_{i}} )

= a^{\binom{r}{2}} ∏_{1≤i<j≤r} (\frac{1/e_{i} − 1/e_{j}}{1/x_{i} − 1/x_{j}} \frac{1}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} \frac{(ax_{i}^{2}q, aq^{2−i}/bc, aq^{2−i}/bd; q)_{∞}}{(ax_{i}q/b, ax_{i}q/c, ax_{i}q/d; q)_{∞}}

× ∏_{i=1}^{r} \frac{(aq/be_{i}, aq^{2−i}/cd, aq/ce_{i}, aq/de_{i}, q, q/ax_{i}^{2}; q)_{∞}}{(ax_{i}q/e_{i}, q/bx_{i}, q/cx_{i}, q/dx_{i}, q/e_{i}x_{i}, a^{2}q^{2−r}/bcde_{i}; q)_{∞}} , (VII.26)

provided the series terminates or |q| < 1 and |a^{2}q^{2−r}/bcde_{i}| < 1 for i = 1, …, r,

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} )

× ∏_{i=1}^{r} \frac{(bx_{i}, cx_{i}, dx_{i}/e_{i}, e_{i}x_{i}; q)_{k_{i}}}{(ax_{i}q/b, ax_{i}q/c, ae_{i}x_{i}q/d, ax_{i}q/e_{i}; q)_{k_{i}}} (\frac{a^{2}q}{bcd} )^{k_{i}} )

= a^{\binom{r}{2}} ∏_{1≤i<j≤r} (\frac{1/e_{i} − 1/e_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − e_{i}e_{j}/d}{1 − ax_{i}x_{j}}) ∏_{i=1}^{r} \frac{(ax_{i}^{2}q, aq^{2−i}/bc, ae_{i}q/bd; q)_{∞}}{(ax_{i}q/b, ax_{i}q/c, ae_{i}x_{i}q/d; q)_{∞}}

× ∏_{i=1}^{r} \frac{(aq/be_{i}, ae_{i}q/cd, aq/ce_{i}, aq/d, q, q/ax_{i}^{2}; q)_{∞}}{(ax_{i}q/e_{i}, q/bx_{i}, q/cx_{i}, e_{i}q/dx_{i}, q/e_{i}x_{i}, a^{2}q^{2−i}/bcd; q)_{∞}} , (VII.27)

provided the series terminates or |q| < 1 and |a^{2}q^{2−r}/bcd| < 1.

Proof. We start with the sum

∑_{k_{1},...,k_{r}=−∞}^{∞} ( ∏_{1≤i<j≤r} (\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} )

× ∏_{i=1}^{r} \frac{(bx_{i}, c_{i}x_{i}, d_{i}x_{i}, e_{i}x_{i}; q)_{k_{i}}}{(ax_{i}q/b, ax_{i}q/c_{i}, ax_{i}q/d_{i}, ax_{i}q/e_{i}; q)_{k_{i}}} (\frac{a^{2}q}{bc_{i}d_{i}e_{i}} )^{k_{i}} ), (VII.28)

and specialize the parameters c_{i}, d_{i} later. We have

∏_{1≤i<j≤r}(\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) = ∏_{1≤i<j≤r}[(1/x_{j}−1/x_{i})(1−ax_{i}x_{j})]^{−1}

× (a/b)^{\binom{r}{2}} q^{−\binom{r}{3}} det_{1≤s,t≤r} ((bq^{−k_{s}}/ax_{s}; q)_{r−t} (bx_{s}q^{k_{s}}; q)_{r−t}) ,

due to the X_{s} ↦ q^{−k_{s}}/x_{s}, A ↦ b/a, B ↦ 0, and C ↦ a case of Lemma A.22.

Hence, using the elementary identities [25, (I.13) and (I.18)]

(bq^{−k_{s}}/ax_{s}; q)_{r−t} =\frac{(b/ax_{s}; q)_{r−t} (ax_{s}q/b; q)_{k_{s}}}{(ax_{s}q^{1−r+t}/b; q)_{k_{s}}} q^{(t−r)k_{s}}

and

(bx_{s}q^{k_{s}}; q)_{r−t} =\frac{(bx_{s}; q)_{r−t} (bx_{s}q^{r−t}; q)_{k_{s}}}{(bx_{s}; q)_{k_{s}}} ,

we may write (VII.28), when multiplied by ∏_{1≤i<j≤r}[(1/x_{j} − 1/x_{i})(1 − ax_{i}x_{j})], as

(a/b)^{\binom{r}{2}} q^{−\binom{r}{3}} det_{1≤s,t≤r} ((b/ax_{s}; q)_{r−t} (bx_{s}; q)_{r−t} ∑_{k_{s}=−∞}^{∞} \frac{1 − ax_{s}^{2}q^{2k_{s}}}{1 − ax_{s}^{2}}

· \frac{(bx_{s}q^{r−t}, c_{s}x_{s}, d_{s}x_{s}, e_{s}x_{s}; q)_{k_{s}}}{(ax_{s}q^{1−r+t}/b, ax_{s}q/c_{s}, ax_{s}q/d_{s}, ax_{s}q/e_{s}; q)_{k_{s}}} (\frac{a^{2}q^{1+t−r}}{bc_{s}d_{s}e_{s}} )^{k_{s}} ).

Now, to the sum inside the determinant we apply Bailey’s classical_{6}ψ_{6} summation, (VII.23), with a ↦ ax_{s}^{2}, b ↦ bx_{s}q^{r−t}, c ↦ c_{s}x_{s}, d ↦ d_{s}x_{s}, and e ↦ e_{s}x_{s}. Thus we obtain

(a/b)^{\binom{r}{2}} q^{−\binom{r}{3}} det_{1≤s,t≤r} ((b/ax_{s}; q)_{r−t} (bx_{s}; q)_{r−t} \frac{(ax_{s}^{2}q, aq^{1−r+t}/bc_{s}; q)_{∞}}{(ax_{s}q^{1−r+t}/b, ax_{s}q/c_{s}; q)_{∞}}

· \frac{(aq^{1−r+t}/bd_{s}, aq^{1−r+t}/be_{s}, aq/c_{s}d_{s}, aq/c_{s}e_{s}, aq/d_{s}e_{s}, q, q/ax_{s}^{2}; q)_{∞}}{(ax_{s}q/d_{s}, ax_{s}q/e_{s}, q^{1−r+t}/bx_{s}, q/c_{s}x_{s}, q/d_{s}x_{s}, q/e_{s}x_{s}, a^{2}q^{1−r+t}/bc_{s}d_{s}e_{s}; q)_{∞}} ).

Now, by using linearity of the determinant with respect to rows, we take some factors out of the determinant and obtain

∏_{i=1}^{r} \frac{(ax_{i}^{2}q, aq/bc_{i}, aq/bd_{i}, aq/be_{i}, aq/c_{i}d_{i}, aq/c_{i}e_{i}, aq/d_{i}e_{i}, q, q/ax_{i}^{2}; q)_{∞}}{(ax_{i}q/b, ax_{i}q/c_{i}, ax_{i}q/d_{i}, ax_{i}q/e_{i}, q/bx_{i}, q/c_{i}x_{i}, q/d_{i}x_{i}, q/e_{i}x_{i}, qa^{2}/bc_{i}d_{i}e_{i}; q)_{∞}}

× (a/b)^{\binom{r}{2}} q^{−\binom{r}{3}} det_{1≤s,t≤r} (\frac{(bc_{s}/a; q)_{r−t} (bd_{s}/a; q)_{r−t} (be_{s}/a; q)_{r−t}}{(bc_{s}d_{s}e_{s}/a^{2}; q)_{r−t}} ) . (VII.29)

To evaluate the determinant in (VII.29) we choose different specializations of the parameters c_{s}, d_{s}, for s = 1, …, r, for which the determinant can be reduced to a product by means of Lemma A.22.

One choice is c_{s} ≡ c, and d_{s} ≡ d. In this case the determinant in (VII.29) equals

∏_{i=1}^{r}[(bc/a; q)_{r−i} (bd/a; q)_{r−i}] det_{1≤s,t≤r} (\frac{(be_{s}/a; q)_{r−t}}{(bcde_{s}/a^{2}; q)_{r−t}} ) .

The determinant can be evaluated by means of Lemma A.22 with X_{s} ↦ e_{s}, A ↦ b/a, B ↦ bcd/a^{2}, and C ↦ 0. Subsequently, substituting our calculations and performing some other elementary manipulations leads to (VII.26).

The other choice is c_{s} ≡ c, and d_{s} ≡ d/e_{s}. In this case the determinant in (VII.29) equals

∏_{i=1}^{r} \frac{(bc/a; q)_{r−i}}{(bcd/a^{2}; q)_{r−i}} det_{1≤s,t≤r} ((bd/ae_{s}; q)_{r−t} (be_{s}/a; q)_{r−t}) .

The determinant can be evaluated by means of Lemma A.22 with X_{s} ↦ e_{s}, A ↦ b/a, B ↦ 0, and C ↦ d. Finally, substituting our calculations and performing some other elementary manipulations leads to (VII.27).

Remark VII.30. Other multivariate extensions, associated to root systems, of the very-well-poised _{6}ψ_{6} summation formula were derived by Gustafson [31] using difference equations. He used these higher-dimensional _{6}ψ_{6} summations to obtain, by specialization and limits, the Macdonald identities for the affine root systems of classical type.

We state our C_{r} extension of Jackson’s balanced very-well poised_{8}ϕ_{7} summation formula.

Theorem VII.31 ((dedicated to Tejasi^{∗}) A C_{r} Jackson’s sum). Let x_{1}, …, x_{r}, and a, b, c, and d be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (VII.32) vanish. Then,

∑_{k_{1},...,k_{r}=0}^{N} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} )

× ∏_{i=1}^{r} \frac{(ax_{i}^{2}, bx_{i}, cx_{i}, dx_{i}, a^{2}x_{i}q^{2−r+N}/bcd, q^{−N}; q)_{k_{i}}}{(q, ax_{i}q/b, ax_{i}q/c, ax_{i}q/d, bcdx_{i}q^{r−1−N}/a, ax_{i}^{2}q^{1+N}; q)_{k_{i}}} · q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{1≤i<j≤r} (\frac{1 − ax_{i}x_{j}q^{N}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} \frac{(ax_{i}^{2}q, aq^{2−i}/bc, aq^{2−i}/bd, aq^{2−i}/cd; q)_{N}}{(aq^{2−r}/bcdx_{i}, ax_{i}q/d, ax_{i}q/c, ax_{i}q/b; q)_{N}} . (VII.32)

Remark VII.33. Note that all summation indices on the left hand side of (VII.32) have the same range. Summing over a cube is strange but essential for the determinant in our derivation of (VII.32) to simplify. In fact, the series

∑_{0≤k_{i}≤N_{i}} ; i=1,2,...,r( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} )

∗Tejasi is the daughter of Gaurav Bhatnagar. She was born on October 27, 1995 in Ohio.

WWW: Tejasi: http://www.math.ohio-state.edu/^{∼}gaurav/index.html

× ∏_{i=1}^{r} \frac{(ax_{i}^{2}, bx_{i}, cx_{i}, dx_{i}, a^{2}x_{i}q^{2−r+N_{i}}/bcd, q^{−N_{i}}; q)_{k_{i}}}{(q, ax_{i}q/b, ax_{i}q/c, ax_{i}q/d, bcdx_{i}q^{r−1−N_{i}}/a, ax_{i}^{2}q^{1+N_{i}}; q)_{k_{i}}} · q^{∑_{i=1}^{r} i k_{i}} )

does not factor, except for N_{1} = N_{2} = ⋯ = N_{r}. To our knowledge, such a phenomenon has not occurred so far with terminating multiple series associated to root systems.

Unfortunately, we cannot use Theorem VII.31 for deriving a multiple_{10}ϕ_{9} transformation, analogously as we derived some new _{10}ϕ_{9} transformations in chapter V.

Proof of Theorem VII.31. We have

q^{∑_{i=1}^{r}(i−r) k_{i}} ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} )

= ∏_{1≤i<j≤r} (\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} )

= ∏_{1≤i<j≤r}[(1/x_{j} − 1/x_{i})(1 − ax_{i}x_{j})]^{−1} ∏_{i=1}^{r} \frac{(q^{2−r−k_{i}}/cx_{i}; q)_{r−1} (ax_{i}q^{2−r+k_{i}}/c; q)_{r−1}}{(aq^{2−r}/bc; q)_{i−1} (bq^{2+r−2i}/c; q)_{i−1}}

× (a/b)^{\binom{r}{2}} q^{−\binom{r}{3}} det_{1≤s,t≤r} (\frac{(bq^{−k_{s}}/ax_{s}; q)_{r−t} (bx_{s}q^{k_{s}}; q)_{r−t}}{(q^{2−r−k_{s}}/cx_{s}; q)_{r−t} (ax_{s}q^{2−r+k_{s}}/c; q)_{r−t}} ) ,

due to the X_{s} ↦ q^{−k_{s}}/x_{s}, A ↦ b/a, B ↦ q^{2−r}/c, and C ↦ a case of Lemma A.22. Hence, using some elementary identities from [25, Appendix I], we may write the left hand side of (VII.32), when multiplied by ∏_{1≤i<j≤r}[(1/x_{j} − 1/x_{i})(1 − ax_{i}x_{j})], as

(a^{2}/bc^{2})^{\binom{r}{2}} q^{−3\binom{r}{3}} ∏_{i=1}^{r}[(aq^{2−r}/bc; q)_{i−1} (bq^{2+r−2i}/c; q)_{i−1}]^{−1}

× det_{1≤s,t≤r} ((b/ax_{s}, bx_{s}; q)_{r−t} (c/ax_{s}, cx_{s}; q)_{t−1} ∑_{k_{s}=0}^{N} \frac{1 − ax_{s}^{2}q^{2k_{s}}}{1 − ax_{s}^{2}}

· \frac{(ax_{s}^{2}, bx_{s}q^{r−t}, cx_{s}q^{t−1}, dx_{s}, a^{2}x_{s}q^{2−r+N}/bcd, q^{−N}; q)_{k_{s}}}{(q, ax_{s}q^{1−r−t}/b, ax_{s}q^{2−t}/c, ax_{s}q/d, bcdx_{s}q^{r−1−N}/a, ax_{i}^{2}q^{1+N}; q)_{k_{s}}} q^{k_{s}} ).

Now, to the sum inside the determinant we apply Jackson’s classical _{8}ϕ_{7} summation, Theorem IV.8, with the replacements a ↦ ax_{s}^{2}, b ↦ bx_{s}q^{r−t}, c ↦ cx_{s}q^{t−1}, d ↦ dx_{s}, and n ↦ N. Thus we obtain

(a^{2}/bc^{2})^{\binom{r}{2}} q^{−3\binom{r}{3}} ∏_{i=1}^{r}[(aq^{2−r}/bc; q)_{i−1} (bq^{2+r−2i}/c; q)_{i−1}]^{−1}

× det_{1≤s,t≤r} ((b/ax_{s}, bx_{s}; q)_{r−t} (c/ax_{s}, cx_{s}; q)_{t−1}

· \frac{(ax_{s}^{2}q, aq^{2−r}/bc, aq^{1−r+t}/bd, aq^{2−t}/cd; q)_{N}}{(aq^{2−r}/bcdx_{s}, ax_{s}q/d, ax_{s}q^{2−t}/c, ax_{s}q^{1−r+t}/b; q)_{N}} ).

Now, we take some factors out of the determinant and obtain

(a^{3}/bc^{4})^{\binom{r}{2}} q^{3N\binom{r}{2}−7\binom{r}{3}} ∏_{i=1}^{r} \frac{(cq^{−N}/ax_{i}; q)_{r−1} (cx_{i}; q)_{r−1}}{(aq^{2−r}/bc; q)_{i−1} (bq^{2+r−2i}/c; q)_{i−1}}

× ∏_{i=1}^{r} \frac{(ax_{i}^{2}q, aq^{2−r}/bc, aq^{1−r+i}/bd, aq^{2−i}/cd; q)_{N}}{(aq^{2−r}/bcdx_{i}, ax_{i}q/d, ax_{i}q/c, ax_{i}q/b; q)_{N}}

× det_{1≤s,t≤r} (\frac{(bq^{−N}/ax_{s}; q)_{r−t} (bx_{s}; q)_{r−t}}{(q^{2−r}/cx_{s}; q)_{r−t} (ax_{s}q^{2−r+N}/c; q)_{r−t}} ) . (VII.34)

The determinant can be evaluated by means of Lemma A.22 with X_{s} ↦ 1/x_{s}, A ↦ bq^{−N}/a, B ↦ q^{2−r}/c, and C ↦ aq^{N}. Subsequently, substituting our calculations and performing some other elementary manipulations leads to (VII.32).

We state an important limiting case of Theorem VII.31, namely an A_{r} extension of Jackson’s [39] q-analog of the Pfaff-Saalschütz formula [25, (1.7.2)], a summation theorem for a terminating and balanced _{3}ϕ_{2} series. The q = 1 case, the 3F_{2} summation theorem [25, (1.7.1)], was originally found by Pfaff [62], and was rediscovered by Saalschütz [65].

Theorem VII.35 (An A_{r} q-Pfaff-Saalschütz sum). Let x_{1}, …, x_{r}, and a, b, and c be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (VII.36) vanish. Then,

∑_{k_{1},...,k_{r}=0}^{N} ( ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} ) ∏_{i=1}^{r} \frac{(ax_{i}, bx_{i}, q^{−N}; q)_{k_{i}}}{(q, cx_{i}, abx_{i}q^{r−N}/c; q)_{k_{i}}} · q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(cq^{1−i}/a, cq^{1−i}/b; q)_{N}}{(cx_{i}, cq^{1−r}/abx_{i}; q)_{N}} . (VII.36)

Proof. The proof is just as in the classical one-dimensional case. Take the limit a → 0 after replacing d by aq/d in (VII.32). Finally, relabel c ↦ a and d ↦ c in the resulting identity to obtain (VII.36).

Other important specializations of our Theorems VII.25 and VII.31 are nonterminating and terminating C_{r} _{6}ϕ_{5} summations.

Theorem VII.37 (C_{r} nonterminating _{6}ϕ_{5} summations). Let x_{1}, …, x_{r}, a, b, c, and d be indeterminate, let r ≥ 1, and suppose that none of the denominators in (VII.38) or (VII.39) vanish. Then

∑_{k_{1},...,k_{r}=0}^{∞} ( ∏_{1≤i<j≤r} (\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} )

× ∏_{i=1}^{r} \frac{(ax_{i}^{2}, bx_{i}, cx_{i}, dx_{i}; q)_{k_{i}}}{(q, ax_{i}q/b, ax_{i}q/c, ax_{i}q/d; q)_{k_{i}}} (\frac{aq}{bcdx_{i}} )^{k_{i}} )

= ∏_{1≤i<j≤r}(1 − ax_{i}x_{j})^{−1} ∏_{i=1}^{r} \frac{(ax_{i}^{2}q, aq^{2−i}/bc, aq^{2−i}/bd, aq^{2−i}/cd; q)_{∞}}{(aq^{2−r}/bcdx_{i}, ax_{i}q/d, ax_{i}q/c, ax_{i}q/b; q)_{∞}} , (VII.38)

provided the series terminates or |q| < 1 and |aq^{2−r}/bcdx_{i}| < 1 for i = 1, …, r,

∑_{k_{1},...,k_{r}=0}^{∞} ( ∏_{1≤i<j≤r} (\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} )

× ∏_{i=1}^{r} \frac{(ax_{i}^{2}, bx_{i}, cx_{i}, d; q)_{k_{i}}}{(q, ax_{i}q/b, ax_{i}q/c, ax_{i}^{2}q/d; q)_{k_{i}}} (\frac{aq}{bcd})^{k_{i}} )

= ∏_{1≤i<j≤r} \frac{(1 − ax_{i}x_{j}/d)}{(1 − ax_{i}x_{j})} ∏_{i=1}^{r} \frac{(ax_{i}^{2}q, aq^{2−i}/bc, ax_{i}q/bd, ax_{i}q/cd; q)_{∞}}{(aq^{2−i}/bcd, ax_{i}^{2}q/d, ax_{i}q/c, ax_{i}q/b; q)_{∞}} , (VII.39)

provided the series terminates or |q| < 1 and |aq^{2−r}/bcd| < 1.

Proof. For (VII.38), let e_{i} ≡ ax_{i} in (VII.26), or, equivalently, N → ∞ in (VII.32). For (VII.39), let e_{i} ≡ ax_{i} and d ↦ ad in (VII.27).

Theorem VII.40 (A C_{r} terminating _{6}ϕ_{5} summation). Let x_{1}, …, x_{r}, a, b, and c be indeterminate, let N be a nonnegative integer, let r ≥ 1, and suppose that none of the denominators in (VII.41) vanish. Then

∑_{k_{1},...,k_{r}=0}^{N} ( ∏_{1≤i<j≤r} (\frac{q^{−k_{i}}/x_{i} − q^{−k_{j}}/x_{j}}{1/x_{i} − 1/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} ) ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} )

× ∏_{i=1}^{r} \frac{(ax_{i}^{2}, bx_{i}, cx_{i}, q^{−N}; q)_{k_{i}}}{(q, ax_{i}q/b, ax_{i}q/c, ax_{i}^{2}q^{1+N}; q)_{k_{i}}} (\frac{aq^{1+N}}{bc} )^{k_{i}} )

= ∏_{1≤i<j≤r} \frac{(1 − ax_{i}x_{j}q^{N})}{(1 − ax_{i}x_{j})} ∏_{i=1}^{r} \frac{(ax_{i}^{2}q, aq^{2−i}/bc; q)_{N}}{(ax_{i}q/b, ax_{i}q/c; q)_{N}} , (VII.41)

provided the series terminates or |q| < 1 and |aq^{2+N−r}/bc| < 1.

Proof. Let d ↦ q^{−N} in (VII.39), or, equivalently, d → ∞ in (VII.32).

If we specialize our above theorems further, we obtain A_{r} extensions of various important classical summation theorems such as the q-Gauß, q-Chu-Vandermonde, and q-binomial theorem.

Finally, we give multiple extensions of Chu’s [18, eq. (15)] general formula transforming a bilateral _{p+2}ψ_{p+2} series of Gasper-Karlsson-Minton type into a multiple of a unilateral (one-sided) _{p+2}ϕ_{p+1} series of Gasper-Karlsson-Minton type. (We say that a _{p+2}ψ_{p+2} series is of Gasper-Karlsson-Minton type if there are p upper parameters a_{1}, …, a_{p} and p lower parameters b_{1}, …, b_{p} such that each a_{i} differs from b_{i} by a nonnegative integer power of q, i.e. a_{i} = b_{i}q^{m_{i}}, m_{i} ≥ 0, [25, sec. 1.9].) Our observation is quite interesting, namely, not only that we may employ several bases q_{1}, …, q_{r} in the multiple series but our calculations can also be carried out using various “Vandermonde determinants”, corresponding to the associated root systems. Further multilateral transformations may be deduced by employing other determinants in our series. Our intention is merely to give an idea how such calculations work.

Chu’s general bilateral transformation formula reads

Theorem VII.42 (Chu). Let a, c, d, z, and b_{1}, …, b_{p} be indeterminate, let N be an integer and m_{1}, …, m_{p} nonnegative integers, and suppose that none of the denominators in (VII.43) vanish. Then

∑_{k=−∞}^{∞} \frac{(a, z, b_{1}q^{m_{1}}, b_{2}q^{m_{2}}, …, b_{p}q^{m_{p}}; q)_{k}}{(c, qdz, b_{1}, b_{2}, …, b_{p}; q)_{k}} (q^{1−N}/a)^{k}

= z^{N} \frac{(qz/a, c/z, qd, q; q)_{∞}}{(qdz, q/z, q/a, c; q)_{∞}} ∏_{j=1}^{p} \frac{(b_{j}/z; q)_{m_{j}}}{(b_{j}; q)_{m_{j}}}

× ∑_{k=0}^{∞} \frac{(1/d, qz/c, qz/b_{1}, qz/b_{2}, …, qz/b_{p}; q)_{k}}{(q, qz/a, q^{1−m_{1}}z/b_{1}, q^{1−m_{2}}z/b_{2}, …, q^{1−m_{p}}z/b_{p}; q)_{k}} (cdq^{N−|m|})^{k} , (VII.43)

provided the series terminate or |q| < 1 and |q/a| < |q^{N}| < |q^{|m|}/cd|. Here, we have used the notation |m| = m_{1} + ⋯ + m_{p}.

Remark VII.44. Note, that when d = 1, the sum on the right hand side of (VII.43) reduces to a single term, and hence, (VII.43) reduces to a summation formula. On the other hand, when c = q we obtain a _{p+2}ϕ_{p+1} transformation (where the N = |m| case was derived in [23]). Jim Haglund [34] has stumbled over the c = q case of (VII.43) via rook theory. He has also noticed that (VII.43) can be obtained by specializing a general transformation formula for bilateral series, due to Slater [70], [25, (5.4.4)].

We give multiple generalizations of (VII.43) associated to the root systems A_{r}, B_{r}, C_{r}, and D_{r} of classical type (cf. [15]).

The determinant evaluations we use in these cases are listed in the following lemma. We remark that the evaluations are basically the Weyl denominator factorizations of type A_{r}, B_{r}, C_{r}, and D_{r}, respectively (cf. [22, Lemma 24.3]).

Lemma VII.45. The following determinant evaluations hold

det_{1≤i,j≤r} (x_{i}^{r−j}) = ∏_{1≤i<j≤r}(x_{i} − x_{j})  (A)

det_{1≤i,j≤r} (x_{i}^{r−j} − x_{i}^{r+j−1}) = ∏_{1≤i<j≤r}[(x_{i} − x_{j})(1 − x_{i}x_{j})] ∏_{i=1}^{r}(1 − x_{i}) (B)

det_{1≤i,j≤r} (x_{i}^{r−j} − x_{i}^{r+j}) = ∏_{1≤i<j≤r}[(x_{i} − x_{j})(1 − x_{i}x_{j})] ∏_{i=1}^{r}(1 − x_{i}^{2}) (C)

\frac{1}{2} · det_{1≤i,j≤r} (x_{i}^{r−j} + x_{i}^{r+j−2}) = ∏_{1≤i<j≤r}[(x_{i} − x_{j})(1 − x_{i}x_{j})].  (D)

Proof. Identities (A), (B), (C), and (D) are readily proved by the standard argument that proves Vandermonde-type determinant evaluations. Namely, first it is verified that the determinant on the left hand side vanishes whenever one of the factors of the right hand side vanishes. Then one checks that both sides are polynomials in x_{1}, …, x_{r} of the same total degree. Hence, the determinant equals a constant times the right hand side. The constant is easily computed by comparing coefficients of a “maximal” or “minimal” monomial.

Let G denote A_{r}, B_{r}, C_{r}, or D_{r}. Besides, let Δ_{G} [x_{1}, …, x_{r}] denote the product side of the G-Vandermonde as displayed in Lemma VII.45. Now we can state a general multilateral transformation formula for basic hypergeometric series associated to G.

Theorem VII.46. Let a_{i}, c_{i}, d_{i}, z_{i}, and b_{i1}, …, b_{ip} be indeterminate, let N_{i} be integers and m_{i1}, …, m_{ip} nonnegative integers, for i = 1, …, r, and suppose that none of the denominators in (VII.47) vanish. Then

∑_{k_{1},...,k_{r}=−∞}^{∞} (\frac{Δ_{G} [q_{1}^{−k_{1}}/x_{1}, …, q_{r}^{−k_{r}}/x_{r}]}{Δ_{G} [1/x_{1}, …, 1/x_{r}]}

× ∏_{i=1}^{r} \frac{(a_{i}, z_{i}, b_{i1}q_{i}^{m_{i1}}, …, b_{ip}q_{i}^{m_{ip}}; q_{i})_{k_{i}}}{(c_{i}, q_{i}d_{i}z_{i}, b_{i1}, …, b_{ip}; q_{i})_{k_{i}}} (q_{i}^{1−N_{i}}/a_{i})^{k_{i}} )

= ∏_{i=1}^{r} z_{i}^{N_{i}} \frac{(q_{i}z_{i}/a_{i}, c_{i}/z_{i}, q_{i}d_{i}, q_{i}; q_{i})_{∞}}{(q_{i}d_{i}z_{i}, q_{i}/z_{i}, q_{i}/a_{i}, c_{i}; q_{i})_{∞}} ∏_{i=1}^{r} ∏_{j=1}^{p} \frac{(b_{ij}/z_{i}; q_{i})_{m_{ij}}}{(b_{ij}; q_{i})_{m_{ij}}}

× \frac{Δ_{G} [z_{1}/x_{1}, …, z_{r}/x_{r}]}{Δ_{G} [1/x_{1}, …, 1/x_{r}]} ∑_{k_{1},...,k_{r}=0}^{∞} (\frac{Δ_{G} [q_{1}^{k_{1}}z_{1}/x_{1}, …, q_{r}^{k_{r}}z_{r}/x_{r}]}{Δ_{G} [z_{1}/x_{1}, …, z_{r}/x_{r}]}

× ∏_{i=1}^{r} \frac{(1/d_{i}, q_{i}z_{i}/c_{i}, q_{i}z_{i}/b_{i1}, …, q_{i}z_{i}/b_{ip}; q_{i})_{k_{i}}}{(q_{i}, q_{i}z_{i}/a_{i}, q_{i}^{1−m_{i1}}z_{i}/b_{i1}, …, q_{i}^{1−m_{ip}}z_{i}/b_{ip}; q_{i})_{k_{i}}} (c_{i}d_{i}q_{i}^{N_{i}−|m_{i}|})^{k_{i}} ), (VII.47)

provided the series terminate or converge (where we have used the notation |m_{i}| = m_{i1} + ⋯ + m_{ip}, for i = 1, …, r).

Sketch of proof. We start with the left hand side of (VII.47). Using Lemma VII.45, we substitute the corresponding determinant for Δ_{G} [q_{1}^{−k_{1}}/x_{1}, …, q_{r}^{−k_{r}}/x_{r}], and by using linearity of the determinant with respect to rows, we write the whole series as a single determinant. Now we transform the terms in the determinant by Theorem VII.42. In the resulting determinant, again by using linearity of the determinant with respect to rows, we take some factors out and are left with a determinant for which we can apply Lemma VII.45 again, obtaining the right hand side of (VII.47). It is an easy exercise to verify that for the respective determinants of Lemma VII.45 the described calculations indeed work out well and yield (VII.47).

Remark VII.48. It is also possible to extend (VII.43) by using other determinants, instead of the respective G-Vandermonde determinants of Lemma VII.45, in the analysis of our above sketch of proof.

For example, we can extend Theorem VII.46 by using the determinantal definition of Schur functions (cf. [22]). Given a partition λ = (λ_{1}, …, λ_{r}) of length ≤ r (cf. [3]), the Schur function S_{λ} is defined by

S_{λ}(x_{1}, …, x_{r}) := \begin{matrix}det_{1≤i,j≤r} & (x_{i}^{r+λ_{j}−j}) \\ det_{1≤i,j≤r} & (x_{i}^{r−j}) \end{matrix}.

We can extend this definition to “Schur functions” associated to root systems of classical type. Given a root system G of classical type, i.e. G is A_{r}, B_{r}, C_{r}, or D_{r}, respectively, and a partition λ = (λ_{1}, …, λ_{r}) of length ≤ r, then define S_{G;λ} (this definition is not standard) by

S_{A_{r};λ}(x_{1}, …, x_{r}) :=S_{λ}(x_{1}, …, x_{r}),  (SA)

S_{B_{r};λ}(x_{1}, …, x_{r}) :=\begin{matrix}det_{1≤i,j≤r} & (x_{i}^{r+λ_{j}−j} − x_{i}^{r+j−λ_{j}−1}) \\ det_{1≤i,j≤r} & (x_{i}^{r−j} − x_{i}^{r+j−1}) \end{matrix},  (SB)

S_{C_{r};λ}(x_{1}, …, x_{r}) :=\begin{matrix}det_{1≤i,j≤r} & (x_{i}^{r+λ_{j}−j} − x_{i}^{r+j−λ_{j}} ) \\ det_{1≤i,j≤r} & (x_{i}^{r−j} − x_{i}^{r+j}) \end{matrix},  (SC)

S_{D_{r};λ}(x_{1}, …, x_{r}) :=\begin{matrix}det_{1≤i,j≤r} & (x_{i}^{r+λ_{j}−j} − x_{i}^{r+j−λ_{j}−2}) \\ det_{1≤i,j≤r} & (x_{i}^{r−j} − x_{i}^{r+j−2}) \end{matrix},  (SD)

compare with Lemma VII.45. With this determinantal definition, we can add an arbitrary partition λ of length ≤ r, in the formulation of Theorem VII.46: It is easily verified that (VII.47) also holds if we replace all G-Vandermonde determinants occurring on both sides in the series of (VII.47) by their multiplication with the corresponding function S_{G;λ},

Δ_{G} [y_{1}, …, y_{r}] ↦ Δ_{G} [y_{1}, …, y_{r}] · S_{G;λ}(y_{1}, …, y_{r}),

for the respective values of y_{1}, …, y_{r}, where all other expressions on both sides of (VII.47) remain unchanged.

98 VII. MORE MULTIDIMENSIONAL SUMMATION THEOREMS

APPENDIX A Some determinant lemmas

Here we provide three determinant lemmas which we needed for proving our theorems. The first two lemmas, which we utilized in the proofs of Theorems III.7 and III.29, are interesting generalizations of the classical Vandermonde determinant evaluation

det_{1≤i,j≤r}(x_{i}^{r−j}) = ∏_{1≤i<j≤r}(x_{i} − x_{j}),

and the “symplectic” Vandermonde determinant evaluation

det_{1≤i,j≤r}(x_{i}^{r−j} − x_{i}^{r+j}) = ∏_{1≤i<j≤r}(x_{i} − x_{j}) ∏_{1≤i≤j≤r}(1 − x_{i}x_{j}),

respectively. The third lemma of this appendix proved to be useful for deriving multiple series identities in chapter VII.

Lemma A.1. Let x_{1}, …, x_{r}, y_{1}, …, y_{r}, a, and c be indeterminate. Then

det_{1≤i,j≤r} (x_{i}^{r+1−j} − a^{r+1−j} \frac{(x_{i} − c/ ∏_{s=1}^{r} y_{s})}{(a − c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{i} − y_{s})}{(a − y_{s})})

= \frac{(a − c/ ∏_{j=1}^{r} x_{j})}{(a − c/ ∏_{j=1}^{r} y_{j})} ∏_{i=1}^{r} \frac{(a − x_{i})}{(a − y_{i})} ∏_{i=1}^{r} x_{i} ∏_{1≤i<j≤r}(x_{i} − x_{j}). (A.2)

Proof. In the determinant on the left side of (A.2) we take x_{i} out of the i-th row, i = 1, …, r, and a^{r−j} out of the j-th column, j = 1, …, r, obtaining

a^{\binom{r}{2}} ∏_{i=1}^{r} x_{i} det_{1≤i,j≤r} ((\frac{x_{i}}{a} )^{r−j} − \frac{a(x_{i} − c/ ∏_{s=1}^{r} y_{s})}{x_{i}(a − c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{i} − y_{s})}{(a − y_{s})}) .

In the last determinant we subtract the r-th column from all other columns. We are left with entries (x_{i}/a)^{r−j} − 1 for i = 1, …, r and j = 1, …, r − 1, but the r-th

100 A. SOME DETERMINANT LEMMAS

column remains unchanged,

1 − \frac{a(x_{i} − c/ ∏_{s=1}^{r} y_{s})}{x_{i}(a − c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{i} − y_{s})}{(a − y_{s})} for i = 1, …, r.

Next we expand the determinant along the last column, to get

∑_{k=1}^{r}(−1)^{r+k} (1 − \frac{a(x_{k} − c/ ∏_{s=1}^{r} y_{s})}{x_{k}(a − c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{k} − y_{s})}{(a − y_{s})}) det_{1≤i≤r,i≠k}_{1≤j≤r−1} ((\frac{x_{i}}{a} )^{r−j} − 1) .

In the minors we take (x_{i}/a − 1) out of the i-th row

det_{1≤i≤r,i≠k}_{1≤j≤r−1} ((\frac{x_{i}}{a} )^{r−j} − 1) = _{∏_{i=1}^{r} ; i≠k} (\frac{x_{i}}{a} − 1) det_{1≤i≤r,i≠k}_{1≤j≤r−1} ( ∑_{s=0}^{r−1−j} (\frac{x_{i}}{a} )^{s}) . (A.3)

Now, the determinant on the right side of (A.3) can be reduced to the Vandermonde determinant

det_{1≤i≤r,i≠k}_{1≤j≤r−1} ((\frac{x_{i}}{a} )^{r−1−j}) ,

and therefore simplifies to

∏_{1≤i<j≤r} ; i,j≠k (\frac{x_{i}}{a} − \frac{x_{j}}{a} ) .

Substituting our calculations, we arrive at

det_{1≤i,j≤r} (x_{i}^{r+1−j} − a^{r+1−j} \frac{(x_{i} − c/ ∏_{s=1}^{r} y_{s})}{(a − c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{i} − y_{s})}{(a − y_{s})})

= ∏_{i=1}^{r}(a − x_{i}) ∏_{i=1}^{r} x_{i} ∏_{1≤i<j≤r}(x_{i} − x_{j})

× ∑_{k=1}^{r} (1 − \frac{a(x_{k} − c/ ∏_{s=1}^{r} y_{s})}{x_{k}(a − c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{k} − y_{s})}{(a − y_{s})}) (a − x_{k})^{−1}_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})^{−1}. (A.4)

We are done if we can show that the sum in (A.4) equals

\frac{(a − c/ ∏_{j=1}^{r} x_{j})}{(a − c/ ∏_{j=1}^{r} y_{j})} \frac{1}{∏_{i=1}^{r}(a − y_{i})}.

This is accomplished by splitting the sum and applying the partial fraction decomposition

∏_{i=1}^{r} \frac{(t − a_{i})}{(t − b_{i})} =1 + ∑_{j=1}^{r} \frac{∏_{i=1}^{r}(b_{j} − a_{i})}{(t − b_{j})_{∏_{i=1}^{r} ; i≠j}(b_{j} − b_{i})},  (A.5)

and the equivalent formula

∏_{i=1}^{r} \frac{(t − a_{i})}{(t − b_{i})} = ∏_{i=1}^{r} \frac{a_{i}}{b_{i}} + ∑_{j=1}^{r} \frac{t ∏_{i=1}^{r}(b_{j} − a_{i})}{(t − b_{j}) b_{j}_{∏_{i=1}^{r} ; i≠j}(b_{j} − b_{i})},  (A.6)

(which can be obtained from (A.5) by the replacements t → 1/t, a_{i} → 1/a_{i}, b_{i} → 1/b_{i}, for i = 1, …, r,) appropriately to its parts. Namely, we write the sum on the right side of (A.4) as

∑_{k=1}^{r} (1 − \frac{a(x_{k} − c/ ∏_{s=1}^{r} y_{s})}{x_{k}(a − c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{k} − y_{s})}{(a − y_{s})}) \frac{1}{(a − x_{k})} \frac{1}{_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})}

= ∑_{k=1}^{r} \frac{1}{(a − x_{k})_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})}

− \frac{a}{(a − c/ ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(a − y_{i})} ∑_{k=1}^{r} \frac{∏_{i=1}^{r}(x_{k} − y_{i})}{(a − x_{k})_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})}

+ \frac{c/ ∏_{j=1}^{r} y_{j}}{(a − c/ ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(a − y_{i})} ∑_{k=1}^{r} \frac{a ∏_{i=1}^{r}(x_{k} − y_{i})}{(a − x_{k})x_{k}_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})}. (A.7)

The first expression on the right hand side of (A.7) can be summed by the partial fraction decomposition (A.6) with a_{i} = 0, t → 1/t, and b_{i} → 1/b_{i}, for i = 1, …, r,

102 A. SOME DETERMINANT LEMMAS

and reduces to

∑_{k=1}^{r} \frac{1}{(a − x_{k})_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})} = \frac{1}{∏_{i=1}^{r}(a − x_{i})},  (A.8)

the second by the partial fraction decomposition (A.5),

\frac{a}{(a − c/ ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(a − y_{i})} ∑_{k=1}^{r} \frac{∏_{i=1}^{r}(x_{k} − y_{i})}{(a − x_{k})_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})}

= \frac{a}{(a − c/ ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(a − y_{i})} (∏_{i=1}^{r} \frac{(a − y_{i})}{(a − x_{i})} − 1) , (A.9)

and the third can be summed by (A.6),

\frac{c/ ∏_{j=1}^{r} y_{j}}{(a − c ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(a − y_{i})} ∑_{k=1}^{r} \frac{a ∏_{i=1}^{r}(x_{k} − y_{i})}{(a − x_{k}) x_{k}_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})}

= \frac{c/ ∏_{j=1}^{r} y_{j}}{(a − c/ ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(a − y_{i})} (∏_{i=1}^{r} \frac{(a − y_{i})}{(a − x_{i})} − ∏_{i=1}^{r} \frac{y_{i}}{x_{i}} ) . (A.10)

Simplifying (A.7) by means of (A.8), (A.9), and (A.10), we get

∑_{k=1}^{r} (1 − \frac{a(x_{k} − c/ ∏_{s=1}^{r} y_{s})}{x_{k}(a − c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{k} − y_{s})}{(a − y_{s})}) (a − x_{k})^{−1}_{∏_{i=1}^{r} ; i≠k}(x_{k} − x_{i})^{−1}

= ∏_{i=1}^{r} \frac{1}{(a − x_{i})} − \frac{a}{(a − c/ ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(a − y_{i})} (∏_{i=1}^{r} \frac{(a − y_{i})}{(a − x_{i})} − 1)

+ \frac{c/ ∏_{j=1}^{r} y_{j}}{(a − c/ ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(a − y_{i})} (∏_{i=1}^{r} \frac{(a − y_{i})}{(a − x_{i})} − ∏_{i=1}^{r} \frac{y_{i}}{x_{i}} )

= \frac{(a − c/ ∏_{j=1}^{r} x_{j})}{(a − c/ ∏_{j=1}^{r} y_{j})} \frac{1}{∏_{i=1}^{r}(a − y_{i})},

which completes the proof of Lemma A.1.

Lemma A.11. Let x_{1}, …, x_{r}, y_{1}, …, y_{r}, and c be indeterminate. Then

det_{1≤i,j≤r} (x_{i}^{r+1−j} − x_{i}^{j} \frac{(x_{i} − c/ ∏_{s=1}^{r} y_{s})}{(1 − x_{i}c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{i} − y_{s})}{(1 − x_{i}y_{s})})

= ∏_{i=1}^{r} \frac{(1 − y_{i}c/ ∏_{j=1}^{r} y_{j})}{(1 − x_{i}c/ ∏_{j=1}^{r} y_{j})} ∏_{i=1}^{r}(1 − x_{i}^{2}) ∏_{i=1}^{r} x_{i}

× ∏_{i,j=1}^{r}(1 − x_{i}y_{j})^{−1} ∏_{1≤i<j≤r}[(x_{i} − x_{j})(1 − x_{i}x_{j})(1 − y_{i}y_{j})]. (A.12)

Proof. Proving (A.12), we will use a completely different method than in the proof of Lemma A.1. In the determinant on the left side of (A.12) we take x_{i}(x_{i}c − ∏_{s=1}^{r} y_{s})^{−1} ∏_{s=1}^{r}(1 − x_{i}y_{s})^{−1} out of the i-th row, i = 1, …, r, obtaining

det_{1≤i,j≤r} (x_{i}^{r+1−j} − x_{i}^{j} \frac{(x_{i} − c/ ∏_{s=1}^{r} y_{s})}{(1 − x_{i}c/ ∏_{s=1}^{r} y_{s})} ∏_{s=1}^{r} \frac{(x_{i} − y_{s})}{(1 − x_{i}y_{s})})

= ∏_{i=1}^{r} \frac{x_{i}}{(x_{i}c − ∏_{j=1}^{r} y_{j})} ∏_{i,j=1}^{r}(1 − x_{i}y_{j})^{−1} · Δ(c, x, y),

where Δ(c, x, y) is the determinant

det_{1≤i,j≤r} (x_{i}^{r−j}(x_{i}c − ∏_{s=1}^{r} y_{s}) ∏_{s=1}^{r}(1 − x_{i}y_{s}) − x_{i}^{j−1}(c − x_{i} ∏_{s=1}^{r} y_{s}) ∏_{s=1}^{r}(x_{i} − y_{s})) .

(A.13)

Thus, in order to establish the lemma, we have to show that

Δ(c, x, y) = ∏_{i=1}^{r}(y_{i}c − ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(1 − x_{i}^{2}) ∏_{1≤i<j≤r}[(x_{i} − x_{j})(1 − x_{i}x_{j})(1 − y_{i}y_{j})].

We will do this by identifying all factors using a polynomial argument.

We see that Δ(c, x, y) is a polynomial in c, x_{i}, y_{i} (i = 1, …, r) of maximal degree (7r^{2} −r)/2. Now observe that if x_{i_{1}} = x_{i_{2}}, for i_{1} ≠ i_{2}, two rows in the determinant (A.13) are equal, hence ∏_{1≤i<j≤r}(x_{i} − x_{j}) must divide Δ(c, x, y). Next suppose x_{i_{1}} = 1/x_{i_{2}} for some i_{1} ≠ i_{2}. In this case the i_{1}-th row is −x_{i_{2}}^{−2r} times the i_{2}-th row which implies that ∏_{1≤i<j≤r}(1 − x_{i}x_{j}) also divides Δ(c, x, y). If x_{i} = 1 or x_{i} = −1 then all entries of the i-th row are zero, so ∏_{i=1}^{r}(1 − x_{i}^{2}) divides Δ(c, x, y).

The remaining factors of Δ(c, x, y) are a bit more delicate to establish. For each special case we will succeed in specifying nontrivial linear combinations of the columns that vanish. Suppose y_{k} = 1/y_{l} for some k ≠ l. Taking −(1 − x_{i}y_{k})(1 − x_{i}/y_{k}) ∏_{s≠k,l} y_{s} out of the i-th row of (A.13), for all i = 1, …, r, we obtain

the determinant

det_{1≤i,j≤r} (x_{i}^{r−j}(1 − x_{i}c/ ∏_{s≠k,l} y_{s}) ∏_{s≠k,l}(1 − x_{i}y_{s}) − x_{i}^{j−1}(x_{i} − c/ ∏_{s≠k,l} y_{s}) ∏_{s≠k,l}(x_{i} − y_{s})) .

(A.14)

We expand the entries of this determinant in terms of the elementary symmetric functions (see [49, p.19])

e_{m}(y_{1}, …, ŷ_{k}, …, ŷ_{l}, …, y_{r}, c/ ∏_{s≠k,l} y_{s}),  (A.15)

of order m with r − 1 arguments, ŷ_{k} and ŷ_{l} indicating that the variables y_{k}, y_{l} are omitted. Namely, if we write e_{m}(y^{(k,l)}) for the elementary symmetric function (A.15) for short, (A.14) can be written as

det_{1≤i,j≤r} ( ∑_{m=0}^{r−1}(−1)^{m}e_{m}(y^{(k,l)}) (x_{i}^{r−j+m} − x_{i}^{j−1+r−1−m})) .  (A.16)

To prove that this determinant vanishes we show that the columns of (A.16) are linearly dependent. As the coefficients for the linear combination we choose (−1)^{j−1}e_{j−1}(y^{(k,l)}), for j = 1, …, r. Then we have

∑_{j=1}^{r}(−1)^{j−1}e_{j−1}(y^{(k,l)}) ∑_{m=0}^{r−1}(−1)^{m}e_{m}(y^{(k,l)}) (x_{i}^{r−j+m} − x_{i}^{j−1+r−1−m})

= ∑_{j=0}^{r−1} ∑_{m=0}^{r−1}(−1)^{j+m}e_{j}(y^{(k,l)})e_{m}(y^{(k,l)}) (x_{i}^{r−j−1+m} − x_{i}^{j+r−1−m}) = 0. (A.17)

That the sum equals 0 is because it is a double sum in j and m with terms that are skew symmetric in j and m. Hence we have proved that ∏_{1≤i<j≤r}(1 − y_{i}y_{j}) divides Δ(c, x, y).

Now suppose c = ∏_{s≠k} y_{s} for some k = 1, …, r. If we take −(1 − x_{i}y_{k})(1 − x_{i}/y_{k}) ∏_{s=1}^{r} y_{s} out of the i-th row of (A.13) for all i = 1, …, r, we obtain the determinant

det_{1≤i,j≤r} (x_{i}^{r−j} ∏_{s≠k}(1 − x_{i}y_{s}) − x_{i}^{j−1} ∏_{s≠k}(x_{i} − y_{s})) .  (A.18)

We expand the entries of this determinant in terms of the elementary symmetric functions

e_{m}(y_{1}, …, ŷ_{k}, …, y_{r}),  (A.19)

of order m with r − 1 arguments, ŷ_{k} indicating that the variable y_{k} is omitted. Namely, if we write e_{m}(y^{(k)}) for the elementary symmetric function (A.19) for short, (A.18) can be written as

det_{1≤i,j≤r} ( ∑_{m=0}^{r−1}(−1)^{m}e_{m}(y^{(k)}) (x_{i}^{r−j+m} − x_{i}^{j−1+r−1−m})) .  (A.20)

To prove that this determinant vanishes we show that the columns of (A.20) are linearly dependent. Here the coefficients (−1)^{j−1}e_{j−1}(y^{(k)}) for j = 1, …, r do the job (compare with (A.17)). Hence ∏_{1≤i≤r}(c − ∏_{s≠i} y_{s}) divides Δ(c, x, y).

Now suppose y_{k} = 0 for some k = 1, …, r. If we take (−x_{i}c) out of the i-th row of (A.13) for all i = 1, …, r, we obtain the determinant (A.18), and we can proceed as above. I.e., we have also shown that ∏_{1≤i≤r} y_{i} divides Δ(c, x, y).

Collecting all factors of Δ(c, x, y) that we have identified so far, we now know that

Δ(c, x, y) = ∏_{i=1}^{r}(y_{i}c − ∏_{j=1}^{r} y_{j}) ∏_{i=1}^{r}(1 − x_{i}^{2})

× ∏_{1≤i<j≤r}[(x_{i} − x_{j})(1 − x_{i}x_{j})(1 − y_{i}y_{j})] · p(c, x, y), (A.21)

where p(c, x, y) is some polynomial in c, x_{i}, y_{i} (i = 1, …, r). But the degree of the factors we already identified amounts to (7r^{2} − r)/2, which is the same degree as that of Δ(c, x, y). Thus the polynomial p has to be a constant which is easily seen to be 1, since the coefficients of c^{0} ∏_{i=1}^{r}(x_{i}^{r−i}y_{i}^{r}) in Δ(c, x, y) and in the product on the right side of (A.21) both equal (−1)^{r}.

Our calculations in the proofs of our multidimensional summation theorems in chapter VII required the following Lemma.

Lemma A.22. Let X_{1}, …, X_{r}, A, B, and C be indeterminate. Then there holds

det_{1≤s,t≤r} (\frac{(AX_{s}; q)_{r−t} (AC/X_{s}; q)_{r−t}}{(BX_{s}; q)_{r−t} (BC/X_{s}; q)_{r−t}} ) = ∏_{1≤i<j≤r}[(X_{j} − X_{i}) (1 − C/X_{i}X_{j})]

× A^{\binom{r}{2}} q^{\binom{r}{3}} ∏_{i=1}^{r} \frac{(B/A; q)_{i−1} (ABCq^{2r−2i}; q)_{i−1}}{(BX_{i}; q)_{r−1} (BC/X_{i}; q)_{r−1}} . (A.23)

This determinant evaluation follows easily from a determinant lemma of Krattenthaler [44, Lemma 34], which we state here without proof.

Let the degree of a Laurent polynomial ∑_{i=M}^{N} a_{i}x^{i}, M, N ∈ ℤ, a_{i} ∈ ℝ and a_{N} ≠ 0, be defined by deg p := N.

Lemma A.24 (Krattenthaler). Let X_{1}, X_{2}, …, X_{r}, A_{2}, A_{3}, …, A_{r}, and C be indeterminates. If p_{0}, p_{1}, …, p_{r−1} are Laurent polynomials with deg p_{j} ≤ j and p_{j}(C/X) = p_{j}(X) for j = 0, 1, …, r − 1, then

det_{1≤s,t≤r} ((A_{r} + X_{s}) ⋯ (A_{t+1} + X_{s})(A_{r} + C/X_{s}) ⋯ (A_{t+1} + C/X_{s}) · p_{t−1}(X_{s}))

= ∏_{1≤i<j≤r}(X_{i} − X_{j})(1 − C/X_{i}X_{j}) ∏_{i=1}^{r} A_{i}^{i−1} ∏_{i=1}^{r} p_{i−1}(−A_{i}), (A.25)

with the convention that empty products (like (A_{r} + X_{s}) ⋯ (A_{t+1} + X_{s}) for t = r) are equal to 1. (The indeterminate A_{1}, which occurs at the right side of (A.25), in fact is superfluous since it occurs in the argument of a constant polynomial.)

Proof of Lemma A.22. By taking some factors out of the determinant, we write the left side of (A.23) as

A^{2\binom{r}{2}} q^{2\binom{r}{3}} ∏_{i=1}^{r} ((BX_{i}; q)_{r−1}^{−1}(BC/X_{i}; q)_{r−1}^{−1})

× det_{1≤s,t≤r} ((1/A − X_{s}) ⋯ (q^{1+t−r}/A − X_{s})(1/A − C/X_{s}) ⋯ (q^{1−r+t}/A − C/X_{s})

· (BX_{s}q^{r−t}; q)_{t−1} (BCq^{r−t}/X_{s}; q)_{t−1}). (A.26)

Now we apply Lemma A.24 with the replacements X_{s} ↦ −X_{s}, A_{t} ↦ q^{t−r}/A, C ↦ C, and p_{j}(X) ↦ (−BXq^{r−1−j}; q)_{j}·(−BCq^{r−1−j}/X; q)_{j}. A few simplifications give the product side of (A.23).

Remark A.27. The special case X_{s} ≡ q^{s} of Lemma A.22 is equivalent to a determinant evaluation which J. A. Wilson [77] utilized to compute biorthogonal rational functions.

APPENDIX B Background information

Here we state a simplification lemma and the multidimensional Jackson’s summations we needed in the proofs of our multiple _{10}ϕ_{9} transformations.

Lemma B.1.

∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − q^{l_{i}−l_{j}}x_{i}/x_{j}} ) ∏_{i,j=1}^{r} \frac{(q^{l_{i}−k_{j}}x_{i}/x_{j}; q)_{k_{i}−l_{i}}}{(q^{1+l_{i}−l_{j}}x_{i}/x_{j}; q)_{k_{i}−l_{i}}}
= (−1)^{|k|−|l|} q^{−\binom{|k|−|l|}{2}} q^{− ∑_{i=1}^{r} i (k_{i}−l_{i})}.

Remark B.2. Lemma B.1 is equivalent to Lemma 4.3 of [55], where it is proved by some elementary manipulations.

Theorem B.3 ((Milne) An A_{r} Jackson’s sum). Let x_{1}, …, x_{r}, a, b, c, and d be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (B.4) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|}}

× ∏_{i=1}^{r} \frac{(dx_{i}; q)_{k_{i}} (a^{2}x_{i}q^{1+|n|}/bcd; q)_{k_{i}}}{(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}}}

× \frac{(b; q)_{|k|} (c; q)_{|k|}}{(aq/d)_{|k|} (bcdq^{−|n|}/a; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} )

= \frac{(aq/bd; q)_{|n|} (aq/cd; q)_{|n|}}{(aq/d; q)_{|n|} (aq/bcd; q)_{|n|}} ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/bc; q)_{n_{i}}}{(ax_{i}q/b; q)_{n_{i}} (ax_{i}q/c; q)_{n_{i}}} . (B.4)

Remark B.5. Theorem B.3 appeared as Theorem 6.14 in [52] with different notation. It is stated almost identical to the above form in [59, A12].

Theorem B.6. ((Denis and Gustafson, Milne and Lilly) A C_{r} Jackson’s sum). Let x_{1}, …, x_{r}, a, b, c, and d be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (B.7) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}^{2}q^{2k_{i}}}{1 − ax_{i}^{2}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} \frac{1 − ax_{i}x_{j}q^{k_{i}+k_{j}}}{1 − ax_{i}x_{j}} )

× ∏_{i=1}^{r} \frac{(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}} (dx_{i}; q)_{k_{i}} (a^{2}x_{i}q^{1+|n|}/bcd; q)_{k_{i}}}{(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}} (bcdx_{i}q^{−|n|}/a; q)_{k_{i}}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}} (ax_{i}x_{j}q^{1+n_{j}}; q)_{k_{i}}} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{1≤i<j≤r}(ax_{i}x_{j}q; q)_{n_{i}+n_{j}}^{−1} ∏_{i,j=1}^{r}(ax_{i}x_{j}q; q)_{n_{i}}

× \frac{(aq/bc; q)_{|n|} (aq/bd; q)_{|n|} (aq/cd; q)_{|n|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{n_{i}} (ax_{i}q/c; q)_{n_{i}} (ax_{i}q/d; q)_{n_{i}} (aq^{1+|n|−n_{i}}/bcdx_{i}; q)_{n_{i}}]}. (B.7)

Remark B.8. Theorem B.6 appeared as Theorem 4.1 in [19] with different notation. In a form very similar to ours above, with relabeled parameters and slightly rearranged product side, it appeared (independently from [19]) as Theorem 6.13 in [58].

Theorem B.9 ((Bhatnagar) A D_{r} Jackson’s sum). Let x_{1}, …, x_{r}, a, b, c, and d be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (B.10) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{k_{i}+k_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{k_{i}}^{−1}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (d/x_{i}; q)_{|k|}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (d/x_{i}; q)_{|k|−k_{i}}}

× \frac{∏_{i=1}^{r} [(bx_{i}; q)_{k_{i}} (cx_{i}; q)_{k_{i}} (a^{2}x_{i}q^{1+|n|}/bcd; q)_{k_{i}}]}{(aq/b; q)_{|k|} (aq/c; q)_{|k|} (bcdq^{−|n|}/a; q)_{|k|}} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{1≤i<j≤r}(ax_{i}x_{j}q/d; q)_{n_{i}+n_{j}} ∏_{i,j=1}^{r}(ax_{i}x_{j}q/d; q)_{n_{i}}^{−1}

× \frac{∏_{i=1}^{r} [(ax_{i}q; q)_{n_{i}} (ax_{i}q/cd; q)_{n_{i}} (ax_{i}q/bd; q)_{n_{i}} (aq^{1+|n|−n_{i}}/bcx_{i}; q)_{n_{i}}]}{(aq/b; q)_{|n|} (aq/c; q)_{|n|} (aq/bcd; q)_{|n|}}. (B.10)

Remark B.11. Theorem B.9 appears in [12] where it has been derived by combining a D_{r} _{3}ϕ_{2} summation and a D_{r} very-well-poised _{6}ϕ_{5} summation theorem.

Theorem B.9 is also equivalent to Theorem B.12 since it follows from reversing the sum in (B.13) and relabelling parameters.

Theorem B.12 (A D_{r} Jackson’s sum). Let x_{1}, …, x_{r}, a, b, c, and d be indeterminate, let n_{1}, …, n_{r} be nonnegative integers with r ≥ 1, and suppose that none of the denominators in (B.13) vanish. Then

∑_{0≤k_{i}≤n_{i}} ; i=1,2,...,r( ∏_{i=1}^{r} (\frac{1 − ax_{i}q^{k_{i}+|k|}}{1 − ax_{i}} ) ∏_{1≤i<j≤r} (\frac{1 − q^{k_{i}−k_{j}}x_{i}/x_{j}}{1 − x_{i}/x_{j}} )

× ∏_{1≤i<j≤r}(a^{2}x_{i}x_{j}q/bcd; q)_{k_{i}+k_{j}}^{−1} ∏_{i,j=1}^{r}(a^{2}x_{i}x_{j}q^{1+n_{j}}/bcd; q)_{k_{i}}

× ∏_{i,j=1}^{r} \frac{(q^{−n_{j}}x_{i}/x_{j}; q)_{k_{i}}}{(qx_{i}/x_{j}; q)_{k_{i}}} ∏_{i=1}^{r} \frac{(ax_{i}; q)_{|k|} (bcd/ax_{i}; q)_{|k|−k_{i}}}{(ax_{i}q^{1+n_{i}}; q)_{|k|} (bcdq^{−n_{i}}/ax_{i}; q)_{|k|}}

× \frac{(b; q)_{|k|} (c; q)_{|k|} (d; q)_{|k|}}{∏_{i=1}^{r} [(ax_{i}q/b; q)_{k_{i}} (ax_{i}q/c; q)_{k_{i}} (ax_{i}q/d; q)_{k_{i}}]} q^{∑_{i=1}^{r} i k_{i}} )

= ∏_{i=1}^{r} \frac{(ax_{i}q; q)_{n_{i}} (ax_{i}q/bc; q)_{n_{i}} (ax_{i}q/bd; q)_{n_{i}} (ax_{i}q/cd; q)_{n_{i}}}{(ax_{i}q/bcd; q)_{n_{i}} (ax_{i}q/d; q)_{n_{i}} (ax_{i}q/c; q)_{n_{i}} (ax_{i}q/b; q)_{n_{i}}} . (B.13)

Remark B.14. Theorem B.12 is just Theorem IV.22 with slightly relabeled parameters.

Theorem B.12 is also equivalent to Theorem B.9 since it follows from reversing the sum in (B.10) and relabelling parameters.

110 B. BACKGROUND INFORMATION

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