Basic Hypergeometric Series (Encyclopedia of Mathematics and by George Gasper

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 + ~) ...

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