Basic Hypergeometric Series, Second Edition (Encyclopedia of by George Gasper, Mizan Rahman PDF

By George Gasper, Mizan Rahman

ISBN-10: 0511526253

ISBN-13: 9780511526251

ISBN-10: 0521833574

ISBN-13: 9780521833578

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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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 [1926]' 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 [1983] 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.

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Basic Hypergeometric Series, Second Edition (Encyclopedia of Mathematics and its Applications) by George Gasper, Mizan Rahman

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