## Basic hypergeometric series by George Gasper By George Gasper

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

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2) which clearly approaches n! as q ---+ 1-. Hence f q(n + 1) tends to f( n + 1) = n! as q ---+ 1-. 1). 3) q---+I we shall give a simple, formal proof due to Gosper; see Andrews . 11]) and the well-known functional equation for the gamma function r(x + 1) = xr(x), r(1) = 1. 4) For a rigorous justification of the above steps see Koornwinder . 5) f(x + 1) = - 1 - f (x), f(l) = 1. 1) is a natural q-analogue of f(x). 1) that fq(x) has poles at x = 0, -1, -2, .... The residue at x = -n is lim (x x---+-n + n) f q(x) (1 _q)n+1 x+n lim (1 - q-n)(1 - ql-n) ...

M r )] r+ 2 'l-'r+ 1 b b b ' q, a q q, 1,···, r = (q, bq/a; q)oo (bl /b; q)ml ... (br/b; q)mr bm1+ ... + mr (bq, q/a; q)oo (b l ; q)ml ... 1). 6) by replacing a, b, bl , ... , br by qa, qb, qb 1, ... , qb r , respectively, and letting q ----7 1. 6) gives '" [a,blqml, ... ,brqmr. -I -cm1+ ... +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.

2). Also see Andrews . 4) can be reversed (by replacing k by n - k) to yield Sears' [1951c] transformation formula 2¢1 = (q-n, b; c; q, z) (~jb; r)n c;q n (bZ)n 3¢2(q-n, qjz, c-1ql-n; bc-1ql-n, 0; q, q). 1). First, set c = bzq! 1) and letting a ---4 00 qn 2 zn ~ (q,zq;q)n 1 (zq; q)oo . , q! 2) (Zq2; q)oo. 8) n= 1 in terms of infinite products. 11) n= 1 and 00 194 (X, q) = II (1- q2n)(1_ 2q2n-1 cos2x + q4n-2). 12) n= 1 It is common to write 19 k (x) for 19 k (x, q), k = 1, ... ,4. 13) one can think of the theta functions 19 1 (x, q) and 192 (x, q) as one-parameter deformations (generalizations) of the trigonometric functions sin x and cos x, 1.