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)
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  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 ...