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

By George Gasper

An effective 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 checklist of approximately 500 references overlaying the total topic is well worth the fee alone.

Lorenz H. Menke, Jr.

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

Example text

M r )] r+ 2 'l-'r+ 1 b b b ' q, a q q, 1,···, r = (q, bq/a; q)oo (bl /b; q)ml ... (br/b; q)mr bm1+ ... + mr (bq, q/a; q)oo (b l ; q)ml ... 1). 6) by replacing a, b, bl , ... , br by qa, qb, qb 1, ... , qb r , respectively, and letting q ----7 1. 6) gives '" [a,blqml, ... ,brqmr. -I -cm1+ ... +mr)] _ 0 I -I -Cm1+···+mr)1 r+ I '/'r bI, ... 7) while letting b ----700 in the case a '" r+I'/'r = q-Cml+ ... +mr) ' bI qml , ... , br qmr 1 bI, ... , br ;q, (_I)ml+···+mr(. 8) bl ; q ml ... 3). 9) when la-Iql-Cm1+···+mr)1 < 1.

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 + ~) ...

2) by replacing a, b, c in (1. 2) by qa, qb, qC, respectively, and letting q -+ 1. 3). 1) 2'f'1 a, ,aq ,q, q (aq jb , _ q jb',q) 00 , which is a q-analogue of Kummer's formula 2FI(a,b;l+a-b;-I)= r(1 ( r + a - b)r(1 + ta) ) ( I ). 9 q-analogues of the Karlsson-Minton summation formulas Minton [1970] showed that if a is a negative integer and ml, m2, ... , mr are nonnegative integers such that -a 2: ml + ... + mT) then F r+ 2 r+ [a, b, I bb ++ 1, b l + mr . 1] ml , ... , br l , ... , br ' r(b + 1)r(1 - a) (b l - b)ml ...

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