By George Gasper

An excellent reference at the topic. fabric on generalized hypergeometric capabilities (starting with Gauss' hypergeometric functionality) is gifted by means of the q analogy's. the fabric is complicated and is easily written with a decent and readable typeface. The creation to q sequence will fulfill the newbie. The record of approximately 500 references protecting the whole topic is definitely worth the rate alone.

Lorenz H. Menke, Jr.

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

**Sample text**

Q; q)m Izl < 1 and Ibl < 1, '" ( b.. 1). 2) has a q-analogue of the form 2

48) f [al,a2, ... [b l , ... , bs]q;n (a1 A.. -_ r'f/s q , qa2 , ... , qar ,• qb, , ... 49). 50) , sin( 7ru) for noninteger values of u and view [a; u] as a trigonometric deformation of a since lima-to [a; u] = a. The corresponding rts trigonometric hypergeometric series can be defined by rts(al, a2, ... , ar; bl , ... , bs ; u, z) = f [al,a2, ... [bl , ... 51 ) Basic hypergeometric series 8 where n [n; u]! 54) where q = e 2nia , it follows that (qa;q)n n(l-a)/2-n(n-I)/4 [a,. 55) rts(al' a2,···, ar ; bl ,···, bs; u, z) rI-.

1) is a natural q-analogue of f(x). 1) that fq(x) has poles at x = 0, -1, -2, .... The residue at x = -n is lim (x x---+-n + n) f q(x) (1 _q)n+1 x+n lim (1 - q-n)(1 - ql-n) ... 6) = --;--------:-----:----'---:--'----,-----------,----:-:- 22 Basic hypergeometric series The q-gamma function has no zeros, so its reciprocal is an entire function with zeros at x = 0, -1, -2, .... 7) n= 0 q the function 1/r q (x) has zeros at x = -n ± 27rik / log q, where k and n are nonnegative integers. r(x)r (x + ~) ...