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 It was pointed out by Ismail in an unpublished preprint in 1999 (rewritten for publication as Ismail [2003c]) that J~3) (x; q) was actually introduced by Jackson [1905a], contrary to the claim in Swarttouw 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