By George Gasper
An exceptional reference at the topic. fabric on generalized hypergeometric features (starting with Gauss' hypergeometric functionality) is gifted through the q analogy's. the cloth is complicated and is definitely written with a decent and readable typeface. The creation to q sequence will fulfill the newbie. The record of approximately 500 references overlaying the whole topic is definitely worth the rate alone.
Lorenz H. Menke, Jr.
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Extra resources for Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications)
Br = qk al a2 ... ar+ 1 and z = q, and a I-balanced series is called balanced (or Saalschutzian). 7, where we derive a summation formula for such a series. 27) 6 Basic hypergeometric series 1 1 (a;q)-n= (1-aq-l)(1-aq-2) ... 28) where n = 0,1, .... 29) k= 0 for Iql < 1. 29) diverges when a -=I- 0 and Iql 2:: 1, whenever (a; q)oo appears in a formula, we shall assume that Iql < 1. 40) = where nand k are integers. A more complete list of useful identities is given in Appendix I at the end of the book.
Derive the generating function n f:'~o Hn(xlq) (q; q)n t 1 = (teiIJ, te-iIJ; q)oo' It I < 1. 29 The continuous q-ultraspherical polynomials are defined in Askey and Ismail  by C ( . J n=O n = ((3te iIJ , (3te- iIJ ; q)oo ( ·IJ -"IJ) te' te ' . q ' ,00 ,It I < 1. 31 Let ~b ml , ... ,mr are nonnegative integers, then denote the q-difference operator defined for a fixed q by ~bJ(Z) Then ~1 is the ~ = bJ(qz) - J(z). 20). Show that ~bXn = (bqn - 1)xn and, if vn () z then (al,"" ar; q)n (_1)(1+ s-r)n q(l+ s-r)n(n-l)/2 z n, = (q,b 1 ,···,bs;q )n (~~bdq~b2/q'" ~bs/q)Vn(z) = z(~al ~a2 ...
Br)m r where, as usual, it is assumed that none of the factors in the denominators of the terms of the series is zero. , if Re( -a) > ml + ... 1) that F [a, r+1 r b+b l ml , ... , br b 1, ... , r + mr , . 1] - 0 - , Re (-a) > ml + ... 3) These formulas are particularly useful for evaluating sums that appear as solutions to some problems in theoretical physics such as the Racah coefficients. They were also used by Gasper [1981b] to prove the orthogonality on (0,27f) of certain functions that arose in Greiner's  work on spherical harmonics on the Heisenberg group.