By George Gasper, Mizan Rahman
This up-to-date variation will proceed to fulfill the desires for an authoritative entire research of the speedily transforming into box of simple hypergeometric sequence, or q-series. It comprises deductive proofs, routines, and beneficial appendices. 3 new chapters were additional to this version masking q-series in and extra variables: linear- and bilinear-generating features for easy orthogonal polynomials; and summation and transformation formulation for elliptic hypergeometric sequence. moreover, the textual content and bibliography were multiplied to mirror fresh advancements. First version Hb (1990): 0-521-35049-2
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Extra resources for Basic Hypergeometric Series, Second Edition (Encyclopedia of Mathematics and its Applications)
Mr)] _ 0 I -I -Cm1+···+mr)1 r+ I '/'r bI, ... 7) while letting b ----700 in the case a '" r+I'/'r = q-Cml+ ... +mr) ' bI qml , ... , br qmr 1 bI, ... , br ;q, (_I)ml+···+mr(. 8) bl ; q ml ... 3). 9) when la-Iql-Cm1+···+mr)1 < 1. 9) to obtain n 2 ml '" r+ I '/'r [ + ... + mn mr. ] - 0 q -n , bI qml , ... , br q bI, ... 8) '" [q-n,blqml, ... ,brqmr. 12) where n 2 ml + ... 10). 11) when n > ml + ... + mn and it is the a = q-Cml+ ... 9) when n = ml + ... +mr . 10 The q-gamma and q- beta functions 21 was introduced by Thomae  and later by Jackson [1904e].
_ . 3 The q-binomial theorem 9 which was derived by Cauchy ' Heine  and by other mathematicians. See Askey [1980a], which also cites the books by Rothe  and Schweins ' and the remark on p. 491 of Andrews, Askey, and Roy  concerning the terminating form of the q-binomial theorem in Rothe . 2), which can also be found in the books Heine , Bailey [1935, p. 66] and Slater [1966, p. 2). Let us set .. ( ) Ja Z = ~ (a)n n ~ ,z. n=O n. 3) Since this series is uniformly convergent in Izl ::; differentiate it termwise to get f =f f~(z) = E when 0 < E < 1, we may n(a/ n zn-l n= 1 n.
Bs ; z) == rFs [alb' a2,·· ·b' ar ; z] I, ... 19) where a dash is used to indicate the absence of either numerator (when r = 0) or denominator (when s = 0) parameters. 21 ) IFI ( -n; a + 1; x ) . n. Generalizing Heine's series, we shall define an r¢s basic hypergeometric series by Lna (x) A. ( A. [aI, a2, ... , a r r'f's al,a2,···,ar ; bI,···, bs;q,z ) -= r'f's b b ;q,z ] = f n= 0 1, ... 22) with (~) = n(n - 1)/2, where q -=I 0 when r > s + 1. 22) it is assumed that the parameters bl , ... , bs are such that the denominator factors in the terms of the series are never zero.