MULTIDIMENSIONAL MATRIX INVERSIONS AND A_{r} AND D_{r} BASIC HYPERGEOMETRIC SERIES

MICHAEL SCHLOSSER^{†}

Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria. E-mail: SCHLOSSE@Pap.Univie.Ac.At

March 18, 1996

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.

1. Introduction

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 [1] discovered that the Bailey transform [3], 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 [13] 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 [9] had found an even more general matrix inversion though without giving any applications.

Gasper and Rahman [10], [25], [11], [12, sec. 3.6] used another 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 inversions aforementioned, was found by Krattenthaler [16] 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

† Supported by the Austrian Science Foundation FWF, grant P10191-MAT

f_{nk} = \frac{∏_{j=k}^{n−1}(a_{j} − c_{k})}{∏_{j=k+1}^{n}(c_{j} − c_{k})},  (1.1)

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})}.  (1.2)

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 and Rahman.

Multidimensional matrix inversions were found by Milne, Lilly and Bhatnagar. 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 [21, Theorem 3.3], [22], [17], [18], which are higher-dimensional generalizations of Andrews’ Bailey transform matrices, were used to derive A_{r} and C_{r} extensions [21], [23] of many of the classical hypergeometric summation and transformation formulas. Bhatnagar and Milne [4, Theorem 5.7], [6, Theorem 3.48] were even able to find an A_{r} extension of Gasper and Rahman’s bibasic hypergeometric matrix inversion. They used a special case of their 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 paper is a multidimensional extension of Krattenthaler’s matrix inverse (see Theorem 3.1). 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 4.1) which is of different type.

In order to prove our matrix inversions in Theorems 3.1 and 4.1 we utilize Krattenthaler’s operator method [15] which we review in section 2. We adapt a main theorem of [15] and add an appropriate multidimensional corollary (see Corollary 2.4).

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 section 5. We combine a special case of Theorem 3.1 and a C_{r} _{8}ϕ_{7} summation theorem of Milne and Lilly [23] to derive a D_{r} _{8}ϕ_{7} summation theorem, which has been derived independently by Bhatnagar [5] using a different method. We also derive A_{r} and D_{r} extensions of a quadratic hypergeometric summation formula of Gessel and Stanton [13]. Finally, we derive a D_{r} extension of a cubic summation formula of Gasper and Rahman [11].

We are sure that 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 will lead to the discovery of many more new identities. This claim is heavily supported by the fact that identities derived in this paper already lead to new C_{r} and D_{r} extensions of Bailey’s very-well-poised _{10}ϕ_{9} transformation [2]. This is ongoing research undertaken jointly with Bhatnagar [7].

Two determinant evaluations, which are elegant generalizations of the classical and “symplectic” Vandermonde determinants, turn out to be crucial for our computations in sections 3 and 4. We decided to give them in a separate appendix.

This work is part of the author’s thesis, being written under the supervision of C. Krattenthaler. The author feels especially indebted to his supervisor who patiently has provided a lot of help and ideas.

2. An operator method for proving matrix inversions

Let F = (f_{nk})_{n,k∈ℤ^{r}} (as before, ℤ denotes the set of integers) be an infinite lowertriangular 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 [15] 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. 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}}. 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)),

where ⟨z^{0}⟩c(z) denotes the coefficient of z^{0} in c(z). 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 [15, Theorem 1].

Lemma 2.1. Let F = (f_{nk})_{n,k∈ℤ^{r}} be an infinite lower-triangular 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,

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). (2.1)

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,

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).

In our applications we will use a corollary of Lemma 2.1 (see Corollary 2.4). 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}.  (2.2)

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},  (2.3)

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

Proposition 2.2. 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,  (2.4)

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,  (2.5)

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,  (2.6)

where the column indices j and l keep their order.

Then the column determinant det^{→}_{1≤i,j≤r}(V_{ij}), as defined in (2.2) or (2.3), 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},  (2.7)

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}  (2.8)

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}),  (2.9)

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 (2.4) and (2.5), we observe that in every term of (2.9) 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 (2.6) we have X_{ij}(α)X_{kl}(α) = X_{kj}(α)X_{il}(α). This important fact lets all terms in (2.9) 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)},  (2.10)

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

panding det^{→}_{1≤i,j≤r}^{(τ)}(V_{ij}) as in (2.9) 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 (2.8).

Remark 2.3. Note that in the determinant of Proposition 2.2, we may not expand along rows because then the cancellation argument does not apply. (For a counterexample, consider the r = 2 case.)

Corollary 2.4. 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 (2.4), (2.5), and (2.6) of Proposition 2.2. 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,  (2.11)
A_{kl}W_{i}^{(c)} = W_{i}^{(c)}A_{kl},  i ≠ k; i, k, l = 1, …, r,  (2.12)
A_{kl}W_{i}^{(a)} = A_{il}W_{k}^{(a)},  i, k, l = 1, …, r.  (2.13)

Moreover the c_{j}(k) are assumed to satisfy (2.1), and det^{→}_{1≤i,j≤r}(V_{ij}) is assumed to be invertible. With the notation of Lemma 2.1, if

∑_{j=1}^{r} c_{j}(k)V_{ij}f_{k}(z) = W_{i}f_{k}(z),  i = 1, …, r,  (2.14)

then

g_{k}(z) = \frac{1}{⟨f_{k}(z), det^{→}(V_{ij}^{∗})h_{k}(z)⟩} det^{→}(V_{ij}^{∗})h_{k}(z),  (2.15)

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,  (2.16)

with h_{k}(z) ≢ 0 for all k ∈ ℤ^{r}.

Proof. Since it follows by Proposition 2.2 that the column determinant det^{→}_{1≤i,j≤r}(V_{ij}) may be expanded along any column, we can apply Cramer’s rule to (2.14) 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)  (2.17)

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

for j = 1, …, r, and is equivalent to (2.16). Notice that, because of (2.11), (2.12), and (2.13), we may apply Proposition 2.2 in (2.17) and shift W_{i}^{∗} to the right. Now apply Lemma 2.1 with V = det^{→}(V_{ij}) and U_{j} = ∑_{i=1}^{r}(−1)^{i+j}V⃗^{(i,j)}W_{i}f_{k}(z).

3. A multidimensional matrix inversion

For convenience, we introduce the notation |n| = n_{1} + n_{2} + ⋯ + n_{r}.

Theorem 3.1. 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 (3.1) or (3.2) 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}))}  (3.1)

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}))}. (3.2)

Remark 3.2. 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 [21], [22], 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 [22], [17], [18]. 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 [21, 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 [4, Theorem 5.7], [6, Theorem 3.48]. Moreover, the r = 1 case is a restatement of Krattenthaler’s matrix inversion (eqs. (1.1) and (1.2)). Due to the fact that Theorem 3.1 covers all known A_{r} matrix inversions (to the author’s knowledge), we view Theorem 3.1 as an A_{r} matrix inversion theorem (also see Remark 4.2).

Another important special case of Theorem 3.1 is a new multidimensional bibasic hypergeometric matrix inversion, stated seperately as Theorem 5.4 in section 5, which we utilize in our applications.

Proof of Theorem 3.1. We will use the operator method of section 2. From (3.1) 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}, (3.3)

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 (3.3) 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), (3.4)

valid for all k ∈ ℤ^{r}. We want to write our system of equations in a way such that Corollary 2.4 is applicable. In order to achieve this, we expand the products on both sides of (3.4) in terms of the elementary symmetric functions (see [19, 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. (3.5)

Now (3.5) is a system of type (2.14) 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 (2.4), (2.5), (2.6), (2.11), (2.12), and (2.13), the functions c_{j}(k) satisfy (2.1). Hence we may apply Corollary 2.4. The dual system (2.16) 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), (3.6)

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 (3.6) 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 (2.15), 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}]  (3.7)

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 2.2 it follows that the determinant (3.7) 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}) . (3.8)

Note that after expanding the column determinants by (2.10) all summands in (3.8) 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/ ∏_{j=1}^{r} c_{j}(k_{j}))}{(a_{|l|} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{j=1}^{r} \frac{(c_{i}(l_{i}) − c_{j}(k_{j}))}{(a_{|l|} − 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}(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/ ∏_{j=1}^{r} c_{j}(k_{j}))}{(a_{|l|} − b/ ∏_{j=1}^{r} c_{j}(k_{j}))} ∏_{j=1}^{r} \frac{(c_{i}(l_{i}) − c_{j}(k_{j}))}{(a_{|l|} − c_{j}(k_{j}))})h_{kl}z^{−l}. (3.9)

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},  (3.10)

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 2.2. Thus ∑_{l≤k} det_{1≤i,j≤r}(v_{ij})h_{kl}z^{−l} can also be transformed into the right side of (3.9).

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 (3.10) 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}). (3.11)

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 (3.11). Thus, equation (2.15) 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), (3.12)

where g_{k}(z) = ∑_{l≤k} g_{kl}z^{−l}. So, extracting the coefficient of z^{−l} in (3.12) we obtain exactly (3.2).

4. Another multidimensional matrix inversion

Theorem 4.1. Let (c_{i}(t_{i}))_{t_{i}∈ℤ}, i = 1, …, r, be arbitrary sequences, b arbitrary, such that none of the denominators in (4.1) or (4.2) 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}))}  (4.1)

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}))}. (4.2)

Remark 4.2. The special case c_{j}(k_{j}) = x_{j}^{−1}q^{−k_{j}} is a C_{r} generalization of Bressoud’s matrix inversion formula [8], as pointed out in [18, 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 [18]. Therefore, we view Theorem 4.1 as a C_{r} matrix inversion theorem.

Proof of Theorem 4.1. Again, we will use the operator method of section 2. From (4.1) 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}, (4.3)

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 (4.3) 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), (4.4)

I being the identity operator, valid for all k ∈ ℤ^{r}. We will write our system of equations in a way such that Corollary 2.4 is applicable. Again, we expand the products on both sides of (4.4) 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} − z_{i}(−C_{i})^{j}]f_{k}(z)

= [z_{i} + bz_{i} − (−C_{i})^{r+1} − b]f_{k}(z), i = 1, …, r. (4.5)

Now (4.5) is a system of type (2.14) with V_{ij} = [(−C_{i})^{r+1−j}−z_{i}(−C_{i})^{j}], W_{i} = [z_{i}+bz_{i}− (−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 (2.4), (2.5), (2.6), (2.11), (2.12), and (2.13), the functions c_{j}(k) satisfy (2.1). Hence we may apply Corollary 2.4. The dual system (2.16) 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} − (−C_{i}^{∗})^{j}z_{i}]h_{k}(z)

= [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)

= (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), (4.6)

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 (4.6) 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 (2.15), we have to compute the action of

det^{→}_{1≤i,j≤r}(V_{ij}^{∗}) = det^{→}_{1≤i,j≤r}[(−C_{i}^{∗})^{r+1−j} − (−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},  (4.7)

where

v_{ij} = (−c_{i}(l_{i}))^{r+1−j} − (−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.2 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 (4.7) 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}). (4.8)

Again, the pairing ⟨f_{k}(z), det^{→}(V_{ij}^{∗})h_{k}(z)⟩ is simply the coefficient of z^{−k} in (4.8). Thus, equation (2.15) 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),  (4.9)

where g_{k}(z) = ∑_{l≤k} g_{kl}z^{−l}. So, extracting the coefficient of z^{−l} in (4.9) we obtain exactly (4.2).

5. Applications to A_{r} and D_{r} basic hypergeometric series

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. [1], [13], [26]). 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}  (5.1)

if and only if

∑_{0≤l≤k} g_{kl}b_{l} = a_{k}.  (5.2)

We expect that applications of our matrix inversions in Theorems 3.1 and 4.1 will lead to many new identities for multidimensional (basic) hypergeometric series. As an illustration, we use special cases of our Theorem 3.1 to derive A_{r} and D_{r} extensions of a terminating quadratic summation of Gessel and Stanton [13], D_{r} extensions of Jackson’s _{8}ϕ_{7} summation [14], and a D_{r} extension of a cubic summation of Gasper and Rahman [11].

We recall the standard definition of the rising q-factorial (cf. [12]). Define

(a; q)_{∞} := ∏_{j≥0}(1 − aq^{j}),

and for any integer k,

(a; q)_{k} := ∏_{j=0}^{k−1}(1 − aq^{j}) = \frac{(a; q)_{∞}}{(aq^{k}; q)_{∞}} .  (5.3)

Theorem 5.1 (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 (5.4) 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}}} . (5.4)

Remark 5.2. 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}} , (5.5)

due to Gessel and Stanton [13, eq. (1.4), q → q^{2}], to which it reduces for r = 1. Many identities like (5.5), involving bases of different powers of q, are known. Hypergeometric series with several bases were extensively studied by Gasper and Rahman [10], [25], [11], [12, sec. 3.8].

Proof of Theorem 5.1. If we substitute c_{i}(t_{i}) ↦ q^{−2t_{i}}/x_{i}, i = 1, …, r, a_{t} ↦ aq^{t}, and b ↦ a/bx_{1} ⋯ x_{n} in Theorem 3.1 (this special case can be also obtained from the inversion [6, Theorem 3.48] of Bhatnagar and Milne) 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 (5.1) 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, taken from [20, Theorem 6.14] (or in more convenient notation [24, Theorem A12]). This implies the inverse relation (5.2) which is easily transformed into (5.4).

It is not hard to see from a polynomial identity argument that Theorem 5.1 implies the following summation theorem.

Theorem 5.3 (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 (5.6) 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}

(5.6)

Proof. First we write the right sides of (5.6) as quotients of infinite products using (5.3). Then by the b = q^{−N} case of Theorem 5.1 it follows that the identity (5.6) holds for c_{j} = q^{−2n_{j}}, j = 1, …, r. By clearing out denominators in (5.6), 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 5.3.

By another specialization of Theorem 3.1 we obtain an interesting bibasic hypergeometric matrix inversion. We use this inversion to derive D_{r} basic hypergeometric summation formulas. For explanations why we associate D_{r} with these formulas the reader is referred to [5].

Theorem 5.4. 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}  (5.7)

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}. (5.8)

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 3.1 we set 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 we obtain the inverse pair (5.7) and (5.8).

Remark 5.5. The inversion in Theorem 5.4 is a D_{r} extension of Gasper and Rahman’s bibasic matrix inversion [12, (3.6.19) and (3.6.20)], to which it reduces for r = 1.

Theorem 5.6 (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 (5.9) 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}}} . (5.9)

Remark 5.7. For r = 1 Theorem 5.6 reduces to Jackson’s very-well-poised _{8}ϕ_{7} summation formula [14], [12, (II.22)].

Proof of Theorem 5.6. Setting p = q in Theorem 5.4 (i.e. here we consider a D_{r} extension of Bressoud’s matrix inverse [8]) 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 (5.1) 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 Milne and Lilly’s C_{r} _{8}ϕ_{7} summation [23, Theorem 6.13]. This implies the inverse relation (5.2) which is easily transformed into (5.9).

By using a polynomial argument we get

Theorem 5.8 (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 (5.10) 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}} . (5.10)

Proof. First we write the right side of (5.10) as quotient of infinite products using (5.3). Then by the c = q^{−N} case of Theorem 5.6 it follows that the identity (5.10) holds for c_{j} = q^{−n_{j}}, j = 1, …, r. By clearing out denominators in (5.10), 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 5.8.

Limiting cases of Theorem 5.6 or Theorem 5.8 include various D_{r} summations. By reversing the multisum in Theorem 5.6 we obtain another D_{r} Jackson’s sum which was independently derived by G. Bhatnagar [5] using a different method. D_{r} extensions of many of the classical basic hypergeometric summation theorems are given in [5]. Further consequences of the new D_{r} _{8}ϕ_{7} summations, such as C_{r} and D_{r} extensions of Bailey’s very-well-poised _{10}ϕ_{9} transformation formula [2], [12, (III.28)] will be given in [7].

Remark 5.9. We note that (5.9) and (5.10) 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)_{∞}} , (5.11)

provided the series terminates. However, we feel that the forms (5.9) and (5.10) are preferable since the dependence of summation indices in (5.11) (which just hides what is really going on in the sum) is removed.

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

Theorem 5.10 (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 (5.12) 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}}} , (5.12)

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

Remark 5.11. This quadratic summation formula is a D_{r} extension of Gessel and Stanton’s summation [13, eq. (1.4)], displayed in (5.5), to which it reduces for r = 1.

Proof of Theorem 5.10. Doing the replacements q → q^{2}, p → q in Theorem 5.4 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 (5.1) 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 Milne and Lilly’s C_{r} _{8}ϕ_{7} summation [23, Theorem 6.13]. This implies the inverse relation (5.2) which is easily transformed into (5.12).

Remark 5.12. By reversing the multisum in (5.12) we may obtain another, differently looking, extension of Gessel and Stanton’s quadratic summation formula (5.5).

Using the same technique as in the proofs of Theorems 5.3 and 5.8 we obtain from Theorem 5.10 the next two quadratic summation theorems (with minor substitutions of variables in Theorem 5.14).

Theorem 5.13 ( 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 (5.13) 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}

(5.13)

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

Theorem 5.14 (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 (5.14) 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}} , (5.14)

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

Finally, we derive some cubic summations.

Theorem 5.15 (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 (5.15) 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}

× ∏_{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}}} , (5.15)

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

Remark 5.16. 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}} , (5.16)

due to Gasper and Rahman [11, eq. (4.1), c → 1], to which it reduces for r = 1.

Proof of Theorem 5.15. Doing the replacements q → q^{3}, p → q in Theorem 5.4 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 (5.1) 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 Milne and Lilly’s C_{r} _{8}ϕ_{7} summation [23, Theorem 6.13]. This implies the inverse relation (5.2) which is easily transformed into (5.15).

Remark 5.17. By reversing the multisum in (5.15) we may obtain another, differently looking, D_{r} extension of Gasper and Rahman’s cubic summation formula (5.16).

Using the same polynomial argument as above, we derive some more D_{r} cubic summation theorems from Theorem 5.15. For sake of brevity, we write them in compact form as follows.

Theorem 5.18 (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 (5.17) 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})_{∞}} , (5.17)

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

Remark 5.19. The aforementioned D_{r} cubic summation theorems are the cases a^{2} = q^{N}, with N being an arbitrary integer, of Theorem 5.18, where the right side of (5.17) simplifies differently, depending on the sign of N and the residue class of N mod 6.

Appendix A.

Here we provide two determinant lemmas which we needed in the proofs of our Theorems 3.1 and 4.1. Our lemmas 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.

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.1)

Proof. In the determinant on the left side of (A.1) 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 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.2)

Now, the determinant on the right side of (A.2) 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.3)

We are done if we can show that the sum in (A.3) 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.4)

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.5)

(which can be obtained from (A.4) 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.3) 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.6)

The first expression can be summed by the partial fraction decomposition (A.5) with a_{i} = 0, t → 1/t, and b_{i} → 1/b_{i}, for i = 1, …, r, 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.7)

the second by the partial fraction decomposition (A.4),

\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.8)

and the third can be summed by (A.5),

\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.9)

Simplifying (A.6) by means of (A.7), (A.8), and (A.9), 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.2. 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.10)

Proof. Here we use a completely different method than in the proof of Lemma A.1. In the determinant on the left side of (A.10) 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.11)

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.11) 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.11), 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.12)

We expand the entries of this determinant in terms of the elementary symmetric functions (see [19, p.19])

e_{m}(y_{1}, …, ŷ_{k}, …, ŷ_{l}, …, y_{r}, c/ ∏_{s≠k,l} y_{s}),  (A.13)

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.13) for short, (A.12) 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.14)

To prove that this determinant vanishes we show that the columns of (A.14) 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.15)

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.11) 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.16)

We expand the entries of this determinant in terms of the elementary symmetric functions

e_{m}(y_{1}, …, ŷ_{k}, …, y_{r}),  (A.17)

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.17) for short, (A.16) 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.18)

To prove that this determinant vanishes we show that the columns of (A.18) are linearly dependent. Here the coefficients (−1)^{j−1}e_{j−1}(y^{(k)}) for j = 1, …, r do the job (compare with (A.15)). 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.11) for all i = 1, …, r, we obtain the determinant (A.16), 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.19)

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.19) both equal (−1)^{r}.

References

1. G. E. Andrews, “Connection coefficient problems and partitions”, D. Ray-Chaudhuri, ed., Proc. Symp. Pure Math., vol. 34, Amer. Math. Soc., Providence, R. I., 1979, 1–24.

2. W. N. Bailey, “An identity involving Heine’s basic hypergeometric series”, J. London Math. Soc. 4 (1929), 254–257.

3. W. N. Bailey, “Some identities in combinatory analysis”, Proc. London Math. Soc. (2) 49 (1947), 421–435.

4. G. Bhatnagar, Inverse relations, generalized bibasic series and their U(n) extensions, Ph. D. thesis, The Ohio State University, 1995.

5. G. Bhatnagar, “D_{n} basic hypergeometric series”, in preparation.

6. G. Bhatnagar and S. C. Milne, “Generalized bibasic hypergeometric series and their U(n) extensions”, Adv. in Math., to appear.

7. G. Bhatnagar and M. Schlosser, “C_{n} and D_{n} very-well-poised _{10}ϕ_{9} transformations”, in preparation.

8. D. M. Bressoud, “A matrix inverse”, Proc. Amer. Math. Soc. 88 (1983), 446–448.

9. L. Carlitz, “Some inverse relations”, Duke Math. J. 40 (1973), 893–901.

10. G. Gasper, “Summation, transformation and expansion formulas for bibasic series”, Trans. Amer. Soc. 312 (1989), 257-278.

11. G. Gasper and M. Rahman, “An indefinite bibasic summation formula and some quadratic, cubic and quartic summation and transformation formulae”, Canad. J. Math. 42 (1990), 1–27.

12. G. Gasper and M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press, Cambridge, 1990.

13. I. Gessel and D. Stanton, “Application of q-Lagrange inversion to basic hypergeometric series”, Trans. Amer. Math. Soc. 277 (1983), 173–203.

14. F. H. Jackson, “Summation of q-hypergeometric series”, Messenger of Math. 57 (1921), 101–112.

15. C. Krattenthaler, “Operator methods and Lagrange inversion, a unified approach to Lagrange formulas”, Trans. Amer. Math. Soc. 305 (1988), 325–334.

16. C. Krattenthaler, “A new matrix inverse”, Proc. Amer. Math. Soc., to appear.

17. G. M. Lilly, The C_{l} generalization of Bailey’s transform and Bailey’s lemma, Ph. D. Thesis (1991), University of Kentucky.

18. G. M. Lilly and S. C. Milne, “The C_{l} Bailey Transform and Bailey Lemma”, Constr. Approx. 9 (1993), 473–500.

19. I. G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, Oxford University Press, New York/London, 1995.

20. S. C. Milne, “The multidimensional _{1}Ψ_{1} sum and Macdonald identities for A_{l}^{(1)}”, in Theta Functions Bowdoin 1987 (L. Ehrenpreis and R. C. Gunning, eds.), volume 49 (Part 2) of Proc. Sympos. Pure Math., 1989, pp. 323–359.

21. S. C. Milne, “Balanced _{3}ϕ_{2} summation theorems for U(n) basic hypergeometric series”, Adv. in Math., to appear.

22. S. C. Milne and G. M. Lilly, “The A_{l} and C_{l} Bailey transform and lemma”, Bull. Amer. Math. Soc. (N.S.) 26 (1992), 258–263.

23. S. C. Milne and G. M. Lilly, “Consequences of the A_{l} and C_{l} Bailey transform and Bailey lemma”, Discrete Math. 139 (1995), 319–346.

24. S. C. Milne and J. W. Newcomb, “U(n) very-well-poised_{10}ϕ_{9} transformations”, Constr. Approx., to appear.

25. M. Rahman, “Some cubic summation formulas for basic hypergeometric series”, Utilitas Math. 36 (1989), 161–172.

26. J. Riordan, Combinatorial identities, J. Wiley, New York, 1968.

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

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