A Systematic List of Two- and Three-term

Contiguous Relations for Basic Hypergeometric Series

C. Krattenthaler

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

Contiguous relations base on simply two-, three-, four-term, … identities of certain products. Usually these can be derived from special cases of terminating basic hypergeometric summation or transformation formulas. In what follows, the symbols C⟨number⟩ refer to replacements rules of the Mathematica package HYPQ which is

available via WWW at http://www.mat.univie.ac.at/~kratt (cf. [2, 3]).

1. A two-term contiguous relation. This relation results from splitting the first term of a basic hypergeometric series from the rest. Recall that

rϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, z] = ∑_{k=0}^{∞} \frac{(A)_{k}}{(q)_{k}(B)_{k}} ((−1)^{k}q^{\binom{k}{2}})^{1+s−r} z^{k},

where (A) stands for a sequence A_{1}, A_{2}, …, A_{r} and (B) stands for a sequence B_{1}, B_{2}, …, B_{s}. (A)_{k} is understood as an abbreviation for ∏_{i=1}^{r} A_{i}, etc. In the sequel (A) and (B) also may be shorter sequences, however the length will always be clear from the context.

With these conventions we have (C01, C02)

rϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, z] = 1 + \frac{(−1)^{1+s−r}z}{(1 − q)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}q, (qA) \\ q^{2}, (qB) \end{matrix} ; q, q^{1+s−r}z] (1.1)

2. Simple three-term contiguous relations. Take the identity 1 − q^{k} = (1 − q^{k}). When multiplying this identity by z^{k}(A)_{k}/(B)_{k}(q)_{k} and summing over all k, from this identity we get (C11–C13)

rϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}(A) \\ (B) \end{matrix} ; q, qz] = (−1)^{1+s−r}z \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(qA) \\ (qB) \end{matrix} ; q, q^{1+s−r}z]

(2.1)

Next take the identity (1 − aq^{k}) − (1 − a) = a(1 − q^{k}). This gives (C14, C15, C16, [1, Ex. I.9 (ii)])

rϕ_{s}[ \begin{matrix}aq, (A) \\ (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; q, z]

= (−1)^{1+s−r}az \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}aq, (Aq) \\ (Bq) \end{matrix} ; q, q^{1+s−r}z]

(2.2)

when multiplying by z^{k}(aq)_{k−1}(A)_{k}/(q)_{k}(B)_{k} and summing over k.

When multiplying by z^{k}(A)_{k}/(q)_{k}(a)_{k+1}(B)_{k} and summing over k we get (C20– C22, [1, Ex. I.9 (i)])

rϕ_{s}[ \begin{matrix}(A) \\ a, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, (B) \end{matrix} ; q, z]

= \frac{(−1)^{1+s−r}az}{(1 − a)(1 − aq)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(Aq) \\ aq^{2}, (Bq) \end{matrix} ; q, q^{1+s−r}z] (2.3)

Now consider the similar identity (1 − aq^{k}) − (1 − a)q^{k} = 1 − q^{k}. From this we get (C17, C18, C19)

rϕ_{s}[ \begin{matrix}aq, (A) \\ (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ (B) \end{matrix} ; q, qz]

= (−1)^{1+s−r}z \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}aq, (Aq) \\ (Bq) \end{matrix} ; q, q^{1+s−r}z] (2.4)

and (C23–C25)

rϕ_{s}[ \begin{matrix}(A) \\ a, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, (B) \end{matrix} ; q, qz]

= \frac{(−1)^{1+s−r}z}{(1 − a)(1 − aq)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(Aq) \\ aq^{2}, (Bq) \end{matrix} ; q, q^{1+s−r}z] (2.5)

Next consider (1 − aq^{k}) − (1 − bq^{k}) = q^{k}(b − a). Multiplication by (a)_{k}(b)_{k}(A)_{k}/(B)_{k} gives (C26–C27)

(1 − a)_{r}ϕ_{s}[ \begin{matrix}aq, b, (A) \\ (B) \end{matrix} ; q, z] − (1 − b)_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ (B) \end{matrix} ; q, z]

= (b − a)_{r}ϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, qz] . (2.6)

Multiplication by (a)_{k}(A)_{k}/(bq)_{k}(B)_{k} gives (C34–C36)

(1 − a)_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ bq, (B) \end{matrix} ; q, z] − (1 − b)_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z]

= (b − a)_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ bq, (B) \end{matrix} ; q, qz] . (2.7)

Finally, multiplication by (A)_{k}/(aq)_{k}(bq)_{k}(B)_{k} gives (C45–C46)

(1 − a)_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, bq, (B) \end{matrix} ; q, z] − (1 − b)_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b, (B) \end{matrix} ; q, z]

= (b − a)_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, bq, (B) \end{matrix} ; q, z] . (2.8)

Now consider (1 − aq^{k}) b − (1 − bq^{k}) a = b − a. By similar considerations we obtain (C28–C29, [1, Ex. I.10 (i)])

b(1 − a)_{r}ϕ_{s}[ \begin{matrix}aq, b, (A) \\ (B) \end{matrix} ; q, z] − a(1 − b)_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ (B) \end{matrix} ; q, z]

= (b − a)_{r}ϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, z] . (2.9)

and (C37–C39)

b(1 − a)_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ bq, (B) \end{matrix} ; q, z] − a(1 − b)_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z]

= (b − a)_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ bq, (B) \end{matrix} ; q, z] . (2.10)

and (C47–C48)

b(1 − a)_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, bq, (B) \end{matrix} ; q, z] − a(1 − b)_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b, (B) \end{matrix} ; q, z]

= (b − a)_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, bq, (B) \end{matrix} ; q, z] . (2.11)

3. Contiguous relations resulting from the n = 1 case of the q–Saalschütz sum. Set n = 1 in the q–Saalschütz [1, (1.7.2); Appendix (II.12)] sum. The resulting identity is equivalent with

(1 − C)(1 − C/AB) + \frac{C}{AB} (1 − A)(1 − B) = (1 − C/A)(1 − C/B).  (3.1)

Now let us take A = a, B = 1/bq^{k}, C = 1/q^{k}. After some simplification (3.1) reduces to

(1 − aq^{k})(1 − b) − (1 − a)(1 − bq^{k}) = (a − b)(1 − q^{k}).

From this we obtain the contiguous relations (C30–C31, [1, Ex. I.9 (iv)])

rϕ_{s}[ \begin{matrix}aq, b, (A) \\ (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ (B) \end{matrix} ; q, z]

= (−1)^{1+s−r}z(a − b)\frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}aq, bq, (Aq) \\ (Bq) \end{matrix} ; q, q^{1+s−r}z] . (3.2)

when multiplying by z^{k}(a)_{k−1}(b)_{k−1}(A)_{k}/(B)_{k}, and (C40–C42, [1, Ex. I.9 (iii)])

rϕ_{s}[ \begin{matrix}aq, (A) \\ bq, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ b, (B) \end{matrix} ; q, z]

= \frac{(−1)^{1+s−r}z(a − b)}{(1 − b)(1 − bq)} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}aq, (Aq) \\ bq^{2}, (Bq) \end{matrix} ; q, q^{1+s−r}z] . (3.3)

when multiplying by z^{k}(a)_{k−1}(A)_{k}/(b)_{k+1}(B)_{k}, and (C49–C50)

rϕ_{s}[ \begin{matrix}(A) \\ a, bq, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b, (B) \end{matrix} ; q, z]

= \frac{(−1)^{1+s−r}z(a − b)}{(1 − a)(1 − aq)(1 − b)(1 − bq)} \frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}_{r}ϕ_{s}[ \begin{matrix}(Aq) \\ aq^{2}, bq^{2}, (Bq) \end{matrix} ; q, q^{1+s−r}z] . (3.4)

Next in (3.1) set A = a, B = b, C = q^{−k}. Then (3.1) reduces to

(1 − aq^{k})(1 − bq^{k}) − q^{k}(1 − a)(1 − b) = (1 − abq^{k})(1 − q^{k}).

In a similar manner as above, from this we get the contiguous relations (C32–C33, C73)

rϕ_{s}[ \begin{matrix}aq, bq, (A) \\ (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}a, b, (A) \\ (B) \end{matrix} ; q, qz]

= (−1)^{1+s−r}z(1 − abq)\frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}aq, bq, abq^{2}, (Aq) \\ abq, (Bq) \end{matrix} ; q, q^{1+s−r}z] . (3.5)

and (C43–C44, C88, [5, (6.5)])

rϕ_{s}[ \begin{matrix}aq, (A) \\ b, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ bq, (B) \end{matrix} ; q, qz]

= \frac{(−1)^{1+s−r}z(1 − abq)}{(1 − b)(1 − bq)} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}aq, abq^{2}, (Aq) \\ bq^{2}, abq, (Bq) \end{matrix} ; q, q^{1+s−r}z] . (3.6)

and (C51–C52, C98)

rϕ_{s}[ \begin{matrix}(A) \\ a, b, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, bq, (B) \end{matrix} ; q, qz] = \frac{(−1)^{1+s−r}z(1 − abq)}{(1 − a)(1 − aq)(1 − b)(1 − bq)}

\frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}abq^{2}, (Aq) \\ aq^{2}, bq^{2}, abq, (Bq) \end{matrix} ; q, q^{1+s−r}z] . (3.7)

Now in (3.1) we set A = aq^{k}, B = c/b, C = cq^{k}. After some simplification we obtain

(1 − cq^{k})(a − b) + (1 − aq^{k})(b − c) + (1 − bq^{k})(c − a) = 0.

From this identity we obtain the contiguous relations (C53)

rϕ_{s}[ \begin{matrix}a, b, cq, (A) \\ (B) \end{matrix} ; q, z] \frac{(a − b)}{(1 − a)(1 − b)} +_{r}ϕ_{s}[ \begin{matrix}aq, b, c, (A) \\ (B) \end{matrix} ; q, z] \frac{(b − c)}{(1 − b)(1 − c)}

+_{r}ϕ_{s}[ \begin{matrix}a, bq, c, (A) \\ (B) \end{matrix} ; q, z] \frac{(c − a)}{(1 − c)(1 − a)} = 0 (3.8)

and (C54–C55)

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c, (B) \end{matrix} ; q, z] \frac{(a − b)}{(1 − a)(1 − b)} +_{r}ϕ_{s}[ \begin{matrix}aq, b, (A) \\ cq, (B) \end{matrix} ; q, z] \frac{(b − c)}{(1 − b)(1 − c)}

+_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ cq, (B) \end{matrix} ; q, z] \frac{(c − a)}{(1 − c)(1 − a)} = 0 (3.9)

and (C56–C57)

rϕ_{s}[ \begin{matrix}a, (A) \\ bq, c, (B) \end{matrix} ; q, z] \frac{(a − b)}{(1 − a)(1 − b)} +_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ bq, cq, (B) \end{matrix} ; q, z] \frac{(b − c)}{(1 − b)(1 − c)}

+_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ b, cq, (B) \end{matrix} ; q, z] \frac{(c − a)}{(1 − c)(1 − a)} = 0 (3.10)

and (C58)

rϕ_{s}[ \begin{matrix}(A) \\ aq, bq, c, (B) \end{matrix} ; q, z] \frac{(a − b)}{(1 − a)(1 − b)} +_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, bq, cq, (B) \end{matrix} ; q, z] \frac{(b − c)}{(1 − b)(1 − c)}

+_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, b, cq, (B) \end{matrix} ; q, z] \frac{(c − a)}{(1 − c)(1 − a)} = 0 (3.11)

Now we set A = b/a, B = ab/c, C = bq^{k} in (3.1) to finally get (1 − aq^{k})(1 − cq^{k}/a) − (1 − bq^{k})(1 − cq^{k}/b) = q^{k}(b − a)(1 − c/ab).

From this we derive the contiguous relations (C59–C60)

rϕ_{s}[ \begin{matrix}aq, cq/a, b, c/b, (A) \\ (B) \end{matrix} ; q, z] (1 − a)(1 − c/a)

−_{r}ϕ_{s}[ \begin{matrix}a, c/a, bq, cq/b, (A) \\ (B) \end{matrix} ; q, z] (1 − b)(1 − c/b)

= _{r}ϕ_{s}[ \begin{matrix}a, c/a, b, c/b, (A) \\ (B) \end{matrix} ; q, qz] (b − a)(1 − c/ab) (3.12)

and (C64–C66)

rϕ_{s}[ \begin{matrix}aq, cq/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z] (1 − a)(1 − c/a) −_{r}ϕ_{s}[ \begin{matrix}a, c/a, bq, (A) \\ c/b, (B) \end{matrix} ; q, z] (1 − b)(1 − c/b)

= _{r}ϕ_{s}[ \begin{matrix}a, c/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, qz] (b − a)(1 − c/ab) (3.13)

and (C74–C75, [5, (5.4)])

rϕ_{s}[ \begin{matrix}aq, b, (A) \\ c/a, cq/b, (B) \end{matrix} ; q, z] (1 − a)(1 − c/a) −_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ cq/a, c/b, (B) \end{matrix} ; q, z] (1 − b)(1 − c/b)

= _{r}ϕ_{s}[ \begin{matrix}a, b, (A) \\ cq/a, cq/b, (B) \end{matrix} ; q, qz] (b − a)(1 − c/ab) (3.14)

and (C80–C82)

rϕ_{s}[ \begin{matrix}aq, cq/a, (A) \\ bq, cq/b, (B) \end{matrix} ; q, z] (1 − a)(1 − c/a) −_{r}ϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] (1 − b)(1 − c/b)

= _{r}ϕ_{s}[ \begin{matrix}a, c/a, (A) \\ bq, cq/b, (B) \end{matrix} ; q, qz] (b − a)(1 − c/ab) (3.15)

and (C89–C91)

rϕ_{s}[ \begin{matrix}aq, (A) \\ c/a, bq, cq/b, (B) \end{matrix} ; q, z] (1 − a)(1 − c/a)

−_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, b, c/b, (B) \end{matrix} ; q, z] (1 − b)(1 − c/b)

= _{r}ϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, bq, cq/b, (B) \end{matrix} ; q, qz] (b − a)(1 − c/ab) (3.16)

and (C99–C100)

rϕ_{s}[ \begin{matrix}(A) \\ a, c/a, bq, cq/b, (B) \end{matrix} ; q, z] (1 − a)(1 − c/a)

−_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, cq/a, b, c/b, (B) \end{matrix} ; q, z] (1 − b)(1 − c/b)

= _{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, cq/a, bq, cq/b, (B) \end{matrix} ; q, qz] (b − a)(1 − c/ab) (3.17)

Finally, in (3.1) we set A = aq^{k}, B = bq^{k}, C = cq^{2k}. This leads to

(1 − cq^{k}/a)(1 − cq^{k}/b) − \frac{c}{ab}(1 − aq^{k})(1 − bq^{k}) = (1 − c/ab)(1 − cq^{2k}).

From this we obtain the contiguous relations (C61, C110)

rϕ_{s}[ \begin{matrix}a, cq/a, b, cq/b, (A) \\ (B) \end{matrix} ; q, z] (1 − c/a)(1 − c/b)

−_{r}ϕ_{s}[ \begin{matrix}aq, c/a, bq, c/b, (A) \\ (B) \end{matrix} ; q, z] \frac{(1 − a)(1 − b)c}{ab}

= _{r+2}ϕ_{s+2}[ \begin{matrix}q\sqrt{c}, −q\sqrt{c}, a, c/a, b, c/b, (A) \\ \sqrt{c}, −\sqrt{c}, (B) \end{matrix} ; q, z] (1 − c)(1 − c/ab) (3.18)

and (C67–C68, C112)

rϕ_{s}[ \begin{matrix}a, cq/a, b, (A) \\ c/b, (B) \end{matrix} ; q, z] (1 − c/a)(1 − c/b)

−_{r}ϕ_{s}[ \begin{matrix}aq, c/a, bq, (A) \\ cq/b, (B) \end{matrix} ; q, z] \frac{(1 − a)(1 − b)c}{ab}

= _{r+2}ϕ_{s+2}[ \begin{matrix}q\sqrt{c}, −q\sqrt{c}, a, c/a, b, (A) \\ \sqrt{c}, −\sqrt{c}, cq/b, (B) \end{matrix} ; q, z] (1 − c)(1 − c/ab) (3.19)

and (C76–C77, C114, [4, proof of (3.10)])

rϕ_{s}[ \begin{matrix}a, b, (A) \\ c/a, c/b, (B) \end{matrix} ; q, z] (1 − c/a)(1 − c/b)

−_{r}ϕ_{s}[ \begin{matrix}aq, bq, (A) \\ cq/a, cq/b, (B) \end{matrix} ; q, z] \frac{(1 − a)(1 − b)c}{ab}

= _{r+2}ϕ_{s+2}[ \begin{matrix}q\sqrt{c}, −q\sqrt{c}, a, b, (A) \\ \sqrt{c}, −\sqrt{c}, cq/a, cq/b, (B) \end{matrix} ; q, z] (1 − c)(1 − c/ab) (3.20)

and (C83, C115)

rϕ_{s}[ \begin{matrix}a, cq/a, (A) \\ bq, c/b, (B) \end{matrix} ; q, z] (1 − c/a)(1 − c/b)

−_{r}ϕ_{s}[ \begin{matrix}aq, c/a, (A) \\ b, cq/b, (B) \end{matrix} ; q, z] \frac{(1 − a)(1 − b)c}{ab}

= _{r+2}ϕ_{s+2}[ \begin{matrix}q\sqrt{c}, −q\sqrt{c}, a, c/a, (A) \\ \sqrt{c}, −\sqrt{c}, bq, cq/b, (B) \end{matrix} ; q, z] (1 − c)(1 − c/ab) (3.21)

and (C92–C93, C118)

rϕ_{s}[ \begin{matrix}a, (A) \\ c/a, bq, c/b, (B) \end{matrix} ; q, z] (1 − c/a)(1 − c/b)

−_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ cq/a, b, cq/b, (B) \end{matrix} ; q, z] \frac{(1 − a)(1 − b)c}{ab}

= _{r+2}ϕ_{s+2}[ \begin{matrix}q\sqrt{c}, −q\sqrt{c}, a, (A) \\ \sqrt{c}, −\sqrt{c}, cq/a, bq, cq/b, (B) \end{matrix} ; q, z] (1 − c)(1 − c/ab) (3.22)

and (C101, C120)

rϕ_{s}[ \begin{matrix}(A) \\ aq, c/a, bq, c/b, (B) \end{matrix} ; q, z] (1 − c/a)(1 − c/b)

−_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, cq/a, b, cq/b, (B) \end{matrix} ; q, z] \frac{(1 − a)(1 − b)c}{ab}

= _{r+2}ϕ_{s+2}[ \begin{matrix}q\sqrt{c}, −q\sqrt{c}, (A) \\ \sqrt{c}, −\sqrt{c}, aq, cq/a, bq, cq/b, (B) \end{matrix} ; q, z] (1 − c)(1 − c/ab) (3.23)

4. Contiguous relations resulting from the n = 1 case of Jackson’s sum. Set n = 1 in Jackson’s [1, (2.6.2); Appendix (II.22)] _{8}ϕ_{7}-sum. In the resulting identity replace a by CD/q, b by A, c by C/A, d by BD. After some simplification this yields

(1 − A)(1 − C/A)(1 − BD)(1 − CD/B) − (1 − B)(1 − C/B)(1 − AD)(1 − CD/A)

= (B − A)(1 − C/AB)(1 − D)(1 − CD). (4.1)

Now we set A = a, B = b, C = c, and D = q^{k} to get

(1 − a)(1 − c/a)(1 − bq^{k})(1 − cq^{k}/b) − (1 − b)(1 − c/b)(1 − q^{k})(1 − cq^{k}/a)

= (b − a)(1 − c/ab)(1 − q^{k})(1 − cq^{k}).

From this identitiy we derive the contiguous relations (C62, C104)

rϕ_{s}[ \begin{matrix}a, c/a, bq, cq/b, (A) \\ (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, b, c/b, (A) \\ (B) \end{matrix} ; q, z]

= (−1)^{1+s−r}z(b − a)(1 − c/ab)(1 − cq)\frac{∏_{i=1}^{r−4}(1 − A_{i})}{∏_{i=1}^{s}(1 − B_{i})}

r+1ϕ_{s+1}[ \begin{matrix}aq, cq/a, bq, cq/b, cq^{2}, (Aq) \\ cq, (Bq) \end{matrix} ; q, q^{1+s−r}z] (4.2)

and (C69–C70, C105)

rϕ_{s}[ \begin{matrix}a, c/a, bq, (A) \\ c/b, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z]

= (−1)^{1+s−r}z \frac{(b − a)(1 − c/ab)(1 − cq)}{(1 − c/b)(1 − cq/b)} \frac{∏_{i=1}^{r−3}(1 − A_{i})}{∏_{i=1}^{s−1}(1 − B_{i})}

r+1ϕ_{s+1}[ \begin{matrix}aq, cq/a, bq, cq^{2}, (Aq) \\ cq^{2}/b, cq, (Bq) \end{matrix} ; q, q^{1+s−r}z] (4.3)

and (C84–C85, C107)

rϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}aq, cq/a, (A) \\ bq, cq/b, (B) \end{matrix} ; q, z]

= (−1)^{1+s−r}z \frac{(b − a)(1 − c/ab)(1 − cq)}{(1 − b)(1 − bq)(1 − c/b)(1 − cq/b)} \frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}

r+1ϕ_{s+1}[ \begin{matrix}aq, cq/a, cq^{2}, (Aq) \\ bq^{2}, cq^{2}/b, cq, (Bq) \end{matrix} ; q, q^{1+s−r}z] (4.4)

and (C78, C106, [5, (5.6)])

rϕ_{s}[ \begin{matrix}a, bq, (A) \\ cq/a, c/b, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}aq, b, (A) \\ c/a, cq/b, (B) \end{matrix} ; q, z]

= (−1)^{1+s−r}z \frac{(b − a)(1 − c/ab)(1 − cq)}{(1 − c/a)(1 − cq/a)(1 − c/b)(1 − cq/b)} \frac{∏_{i=1}^{r−2}(1 − A_{i})}{∏_{i=1}^{s−2}(1 − B_{i})}

r+1ϕ_{s+1}[ \begin{matrix}aq, bq, cq^{2}, (Aq) \\ cq^{2}/a, cq^{2}/b, cq, (Bq) \end{matrix} ; q, q^{1+s−r}z] (4.5)

and (C94–C95, C108)

rϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, b, c/b, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}aq, (A) \\ c/a, bq, cq/b, (B) \end{matrix} ; q, z]

= (−1)^{1+s−r}z \frac{(b − a)(1 − c/ab)(1 − cq)}{(1 − c/a)(1 − cq/a)(1 − b)(1 − bq)(1 − c/b)(1 − cq/b)} \frac{∏_{i=1}^{r−1}(1 − A_{i})}{∏_{i=1}^{s−3}(1 − B_{i})}

r+1ϕ_{s+1}[ \begin{matrix}aq, cq^{2}, (Aq) \\ cq^{2}/a, bq^{2}, cq^{2}/b, cq, (Bq) \end{matrix} ; q, q^{1+s−r}z] (4.6)

and (C102, C109)

rϕ_{s}[ \begin{matrix}(A) \\ aq, cq/a, b, c/b, (B) \end{matrix} ; q, z] −_{r}ϕ_{s}[ \begin{matrix}(A) \\ a, c/a, bq, cq/b, (B) \end{matrix} ; q, z]

= (−1)^{1+s−r}z \frac{(b − a)(1 − c/ab)(1 − cq)}{(1 − a)(1 − aq)(1 − c/a)(1 − cq/a)(1 − b)(1 − bq)(1 − c/b)(1 − cq/b)}

\frac{∏_{i=1}^{r}(1 − A_{i})}{∏_{i=1}^{s−4}(1 − B_{i})}_{r+1}ϕ_{s+1}[ \begin{matrix}cq^{2}, (Aq) \\ aq^{2}, cq^{2}/a, bq^{2}, cq^{2}/b, cq, (Bq) \end{matrix} ; q, q^{1+s−r}z] (4.7)

Next in (4.1) set A = aq^{k}, B = bq^{k}, C = q^{k}ab/d, D = c/ab. This results into (1 − aq^{k})(1 − cq^{k}/a)(1 − b/d)(1 − c/bd) − (1 − bq^{k})(1 − cq^{k}/b)(1 − a/d)(1 − c/ad)

= \frac{1}{d}(a − b)(1 − c/ab)(1 − q^{k}d)(1 − cq^{k}/d).

From this identity we obtain the contiguous relations (C63, C111)

rϕ_{s}[ \begin{matrix}aq, cq/a, b, c/b, (A) \\ (B) \end{matrix} ; q, z] \frac{(1 − b/d)(1 − c/bd)}{(1 − b)(1 − c/b)}

−_{r}ϕ_{s}[ \begin{matrix}a, c/a, bq, cq/b, (A) \\ (B) \end{matrix} ; q, z] \frac{(1 − a/d)(1 − c/ad)}{(1 − a)(1 − c/a)}

= _{r+2}ϕ_{s+2}[ \begin{matrix}a, c/a, b, c/b, dq, cq/d, (A) \\ d, c/d, (B) \end{matrix} ; q, z] \frac{(a − b)(1 − c/ab)(1 − d)(1 − c/d)}{d(1 − a)(1 − c/a)(1 − b)(1 − c/b)} (4.8)

and (C71–C72, C113)

rϕ_{s}[ \begin{matrix}aq, cq/a, b, (A) \\ cq/b, (B) \end{matrix} ; q, z] \frac{(1 − b/d)(1 − c/bd)}{(1 − b)(1 − c/b)}

−_{r}ϕ_{s}[ \begin{matrix}a, c/a, bq, (A) \\ c/b, (B) \end{matrix} ; q, z] \frac{(1 − a/d)(1 − c/ad)}{(1 − a)(1 − c/a)}

= _{r+2}ϕ_{s+2}[ \begin{matrix}a, c/a, b, dq, cq/d, (A) \\ cq/b, d, c/d, (B) \end{matrix} ; q, z] \frac{(a − b)(1 − c/ab)(1 − d)(1 − c/d)}{d(1 − a)(1 − c/a)(1 − b)(1 − c/b)} (4.9)

and (C86–C87, C117)

rϕ_{s}[ \begin{matrix}aq, cq/a, (A) \\ bq, cq/b, (B) \end{matrix} ; q, z] \frac{(1 − b/d)(1 − c/bd)}{(1 − b)(1 − c/b)}

−_{r}ϕ_{s}[ \begin{matrix}a, c/a, (A) \\ b, c/b, (B) \end{matrix} ; q, z] \frac{(1 − a/d)(1 − c/ad)}{(1 − a)(1 − c/a)}

= _{r+2}ϕ_{s+2}[ \begin{matrix}a, c/a, dq, cq/d, (A) \\ bq, cq/b, d, c/d, (B) \end{matrix} ; q, z] \frac{(a − b)(1 − c/ab)(1 − d)(1 − c/d)}{d(1 − a)(1 − c/a)(1 − b)(1 − c/b)} (4.10)

and (C79, C116)

rϕ_{s}[ \begin{matrix}aq, b, (A) \\ c/a, cq/b, (B) \end{matrix} ; q, z] \frac{(1 − b/d)(1 − c/bd)}{(1 − b)(1 − c/b)}

−_{r}ϕ_{s}[ \begin{matrix}a, bq, (A) \\ cq/a, c/b, (B) \end{matrix} ; q, z] \frac{(1 − a/d)(1 − c/ad)}{(1 − a)(1 − c/a)}

= _{r+2}ϕ_{s+2}[ \begin{matrix}a, b, dq, cq/d, (A) \\ cq/a, cq/b, d, c/d, (B) \end{matrix} ; q, z] \frac{(a − b)(1 − c/ab)(1 − d)(1 − c/d)}{d(1 − a)(1 − c/a)(1 − b)(1 − c/b)} (4.11)

and (C96–C97, C119)

rϕ_{s}[ \begin{matrix}aq, (A) \\ c/a, bq, cq/b, (B) \end{matrix} ; q, z] \frac{(1 − b/d)(1 − c/bd)}{(1 − b)(1 − c/b)}

−_{r}ϕ_{s}[ \begin{matrix}a, (A) \\ cq/a, b, c/b, (B) \end{matrix} ; q, z] \frac{(1 − a/d)(1 − c/ad)}{(1 − a)(1 − c/a)}

= _{r+2}ϕ_{s+2}[ \begin{matrix}a, dq, cq/d, (A) \\ cq/a, bq, cq/b, d, c/d, (B) \end{matrix} ; q, z] \frac{(a − b)(1 − c/ab)(1 − d)(1 − c/d)}{d(1 − a)(1 − c/a)(1 − b)(1 − c/b)} (4.12)

and (C103, C121)

rϕ_{s}[ \begin{matrix}(A) \\ a, c/a, bq, cq/b, (B) \end{matrix} ; q, z] \frac{(1 − b/d)(1 − c/bd)}{(1 − b)(1 − c/b)}

−_{r}ϕ_{s}[ \begin{matrix}(A) \\ aq, cq/a, b, c/b, (B) \end{matrix} ; q, z] \frac{(1 − a/d)(1 − c/ad)}{(1 − a)(1 − c/a)}

= _{r+2}ϕ_{s+2}[ \begin{matrix}dq, cq/d, (A) \\ aq, cq/a, bq, cq/b, d, c/d, (B) \end{matrix} ; q, z] \frac{(a − b)(1 − c/ab)(1 − d)(1 − c/d)}{d(1 − a)(1 − c/a)(1 − b)(1 − c/b)} (4.13)

References

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

2. C. Krattenthaler, HYP and HYPQ — Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively q-binomial sums and basic hypergeometric series, J. Symbol. Comput. 20 (1995), 737–744.

3. C. Krattenthaler, HYPQ, Manual for a Mathematica package for handling basic hypergeometric series, 2002.

4. C. Krattenthaler, A new matrix inverse, Proc. Amer. Math. Soc. 124 (1996), 47–59.

5. C. Krattenthaler and H. M. Srivastava, Summations for basic hypergeometric series involving a q-analogue of the digamma function I, Comput. Math. Appl. 32 (1996), 73–91.