Download Basic hypergeometric series by George Gasper PDF

By George Gasper

This up to date variation will proceed to fulfill the desires for an authoritative entire research of the quickly growing to be box of uncomplicated hypergeometric sequence, or q-series. It contains deductive proofs, workouts, and worthy appendices. 3 new chapters were further to this version overlaying q-series in and extra variables; linear- and bilinear-generating features for easy orthogonal polynomials; and summation and transformation formulation for elliptic hypergeometric sequence. additionally, the textual content and bibliography were multiplied to mirror contemporary advancements. First version Hb (1990): 0-521-35049-2

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L-q with Z = cose. 38 Extend Jacobi's triple product identity to the transformation formula 1+ f)-ltq(~)(an+bn) = (q,a,b;q)=f (ab/q;q)znqn. n=O (q,a,b,ab;q)n n=! Deduce that = ( ) n 2n 2 1+ 2~ ~a q = q; q n= ! ) ~ -a, q 2nq n = aq; q2) = n= ~ ( . ) ( . 2) . S. Chihara [1978], Henrici [1974], Luke [1969], Miller [1968], Nikiforov and Uvarov [1988], Vilenkin [1968], and Watson [1952]. Some techniques for using symbolic computer algebraic systems such as Mathematica, Maple, and Macsyma to derive formulas containing hypergeometric and basic hypergeometric series are discussed in Gasper [1990].

10 Denoting 2¢1 (a, b; c; q, z), 2¢1 (aq±l, b; c, q, z), 2¢1 (a, bq±l; c; q, z) and 2¢1 (a, b; cq±l; q, z) by ¢, ¢(aq±I), ¢(bq±l) and ¢(cq±I), respectively, show that (i) b(l - a)¢(aq) - a(l - b)¢(bq) = (b - a)¢, (ii) a(l - b/c)¢(bq-l) - b(l - a/c)¢(aq-l) = (a - b)(l - abz/cq)¢, (iii) q(l - a/c)¢(aq-l) + (1 - a)(l - abz/c)¢(aq) = [1 + q - a - aq/c + a2z(l - b/a)/c]¢, (iv) (1 - c)(q - c) (abz - c)¢(cq-l) + (c - a)(c - b)z¢(cq) = (c - l)[c(q - c) + (ca + cb - ab - abq)z]¢. 11 Let g(e;A,/-l,v) = (AeiO,/-lV;q)oo 2¢1(/-le- iO ,ve- iO ;/-lv;q, Ae iO ).

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 . 15) hm [a; CT, T] = . ( ) = [a;CT]. III T-tOO sIn 1[CT ° Hence, the elliptic number [a; CT, T] is a one-parameter deformation of the trigonometric number [a; CT] and a two-parameter deformation of the number a. Notice that [a; CT, T] is called an "elliptic number" even though it is not an elliptic (doubly periodic and meromorphic) function of a. 5]), any (doubly periodic meromorphic) elliptic function can be written as a constant multiple of a quotient of products of f)l functions.

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