By George Gasper, Mizan Rahman
This up to date version will proceed to fulfill the desires for an authoritative finished research of the quickly growing to be box of uncomplicated hypergeometric sequence, or q-series. It comprises deductive proofs, workouts, and beneficial appendices. 3 new chapters were further to this variation overlaying q-series in and extra variables: linear- and bilinear-generating capabilities for uncomplicated orthogonal polynomials; and summation and transformation formulation for elliptic hypergeometric sequence. furthermore, the textual content and bibliography were extended to mirror fresh advancements. First version Hb (1990): 0-521-35049-2
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Additional resources for Basic Hypergeometric Series, Second Edition (Encyclopedia of Mathematics and its Applications)
2) q ,q ,q 00 q 1/( 1/-1) . 27 Show that 00 n= -CX) 00 Both of these are q-analogues of the generating function 00 L n= t n In(x) = ex (t-C 1 )/2. J( )( ) e , k= 0 q; q k q; q n-k where x = cos B; see Szego ' Carlitz [1955, 1957a, 1958, 1960] and Rogers [1894, 1917]. Derive the generating function n f:'~o Hn(xlq) (q; q)n t 1 = (teiIJ, te-iIJ; q)oo' It I < 1. 29 The continuous q-ultraspherical polynomials are defined in Askey and Ismail  by C ( . J n=O n = ((3te iIJ , (3te- iIJ ; q)oo ( ·IJ -"IJ) te' te ' .
2) (Zq2; q)oo. 8) n= 1 in terms of infinite products. 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 .
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. ;q)n' (a; q)n(qja; q)-n = (-atq(~), (q, -q, _q2; q2)cx; = 1. 2 The q-binomial coefficient is defined by [n] (q;q)n k q - (q; q)k(q; q)n-k for k = 0,1, ... 3+ I; q)cx; (q, q<>+ I; q)cx; for complex a and f3 when Iql (i) (ii) (iii) (q<>+ I; q)k (q;q)k ' < 1.
Basic Hypergeometric Series, Second Edition (Encyclopedia of Mathematics and its Applications) by George Gasper, Mizan Rahman