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

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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Basic Hypergeometric Series, Second Edition (Encyclopedia of Mathematics and its Applications)

This up to date version will proceed to satisfy the desires for an authoritative finished research of the swiftly becoming box of easy hypergeometric sequence, or q-series. It comprises deductive proofs, routines, and worthwhile appendices. 3 new chapters were extra to this variation overlaying q-series in and extra variables: linear- and bilinear-generating capabilities for easy orthogonal polynomials; and summation and transformation formulation for elliptic hypergeometric sequence.

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

Example text

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 [1869] and later by Jackson [1904e].

_ . 3 The q-binomial theorem 9 which was derived by Cauchy [1843]' Heine [1847] and by other mathematicians. See Askey [1980a], which also cites the books by Rothe [1811] and Schweins [1820]' and the remark on p. 491 of Andrews, Askey, and Roy [1999] concerning the terminating form of the q-binomial theorem in Rothe [1811]. 2), which can also be found in the books Heine [1878], 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.

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