By George Gasper
An effective reference at the topic. fabric on generalized hypergeometric features (starting with Gauss' hypergeometric functionality) is gifted through the q analogy's. the cloth is complex and is easily written with a good and readable typeface. The creation to q sequence will fulfill the newbie. The checklist of approximately 500 references overlaying the total topic is definitely worth the rate alone.
Lorenz H. Menke, Jr.
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Additional resources for Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications)
11) n= 1 and 00 194 (X, q) = II (1- q2n)(1_ 2q2n-1 cos2x + q4n-2). 12) n= 1 It is common to write 19 k (x) for 19 k (x, q), k = 1, ... ,4. 13) one can think of the theta functions 19 1 (x, q) and 192 (x, q) as one-parameter deformations (generalizations) of the trigonometric functions sin x and cos x, 1. 7 q-Saalschutz formula 17 respectively. 14) is never zero. 9) it is clear that [a; CT, T] is well-defined, [-a; CT, T] = -[a;CT,TJ, [1;CT,T] = 1, and . 15) hm [a; CT, T] = . ( ) = [a;CT]. III T-tOO sIn 1[CT ° Hence, the elliptic number [a; CT, T] is a one-parameter deformation of the trigonometric number [a; CT] and a two-parameter deformation of the number a.
14) that 1 f is suitably t x-I ((tq; q)oo ) dqt, R ex> 0 , y r--I- 0 ,- 1,- 2, ... , tqY;qoo which clearly approaches the beta function integral B q(x, y ) = 1 o B(x, y) = 11 t x - 1 (1 - t)y-l dt, Re x, Re y > 0, (1117) .. 8) 24 Basic hypergeometric series as q ----+ 1-. 1) in the q-integral form 2¢1 ) _ fq(c) ( a b. c. 10) where Iarg(l- z)1 < 7r and Re c> Re b > O. The q-integral notation is, as we shall see later, quite useful in simplifying and manipulating various formulas involving sums of series.
9). These formulas can be used to derive other important formulas such as, for example, Jackson's [1910a] transformation formula A. ( 2 '/'1 b.. ) = (az; q)oo ~ (a, c/b; q)k (-b )k (~) a, ,e, q, z ( ) ~ ( ) z q z; q 00 k= 0 q, e, az; q k = (az; ( q)oo ) z;q 2¢2 ( a,e/ b;c,az;q,bz ) . 2) to write (a, e - b; e; z/(z - 1)). 2). Also see Andrews . 4) can be reversed (by replacing k by n - k) to yield Sears' [1951c] transformation formula 2¢1 = (q-n, b; c; q, z) (~jb; r)n c;q n (bZ)n 3¢2(q-n, qjz, c-1ql-n; bc-1ql-n, 0; q, q).
Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) by George Gasper