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.