By George Gasper, Mizan Rahman

ISBN-10: 0511526253

ISBN-13: 9780511526251

ISBN-10: 0521833574

ISBN-13: 9780521833578

This up-to-date variation will proceed to satisfy the desires for an authoritative complete research of the swiftly starting to be box of easy hypergeometric sequence, or q-series. It comprises deductive proofs, workouts, and necessary appendices. 3 new chapters were additional to this version masking q-series in and extra variables: linear- and bilinear-generating features for uncomplicated orthogonal polynomials; and summation and transformation formulation for elliptic hypergeometric sequence. furthermore, the textual content and bibliography were increased to mirror contemporary advancements. First variation Hb (1990): 0-521-35049-2

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**Extra resources for Basic Hypergeometric Series, Second Edition (Encyclopedia of Mathematics and its Applications)**

**Sample text**

A different characterization of fq is presented in Kairies and Muldoon [1982]. 27 Other formulas involving q-Bessel functions are contained in Jackson [1904a-d, 1908], Ismail and Muldoon [1988], Rahman [1987, 1988c, 1989b,c], and Swarttouw and Meijer [1994]. It was pointed out by Ismail in an unpublished preprint in 1999 (rewritten for publication as Ismail [2003c]) that J~3) (x; q) was actually introduced by Jackson [1905a], contrary to the claim in Swarttouw [1992] that a special case of it was first discovered by Hahn [1953] and then in full generality by Exton [1978].

Br)m r where, as usual, it is assumed that none of the factors in the denominators of the terms of the series is zero. , if Re( -a) > ml + ... 1) that F [a, r+1 r b+b l ml , ... , br b 1, ... , r + mr , . 1] - 0 - , Re (-a) > ml + ... 3) These formulas are particularly useful for evaluating sums that appear as solutions to some problems in theoretical physics such as the Racah coefficients. They were also used by Gasper [1981b] to prove the orthogonality on (0,27f) of certain functions that arose in Greiner's [1980] work on spherical harmonics on the Heisenberg group.

4) The bilateral q-integral is defined by 1 00 f(t) dqt = (1 - q) f n= -00 [f(qn) + f( _qn)] qn. 5) when restricted. 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.

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

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