Download Basic Hypergeometric Series (Encyclopedia of Mathematics and by George Gasper PDF

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.

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Extra resources for Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications)

Example text

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 [1983] 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 [1980] work on spherical harmonics on the Heisenberg group.

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