A new look at q-hypergeometric functions

Abstract

For complex parameters ai, bj , q(i = 1, ..., r, j = 1, ..., s, bj ∈ C\{0, −1, −2, ...}, |q| < 1), define the q-hypergeometric function rΦs(a1, ..., ar; b1, ..., bs; q, z) by rΦs(ai; bj ; q, z) = ∑∞ n=0 (a1, q)n...(ar, q)n (q, q)n(b1, q)n...(bs, q)n z n (r = s + 1; r, s ∈ N0 = N ∪ {0}; z ∈ U) where N denote the set of positive integers and (a, q)n is the q-shifted factorial defined by (a, q)n = { 1, n = 0; (1 − a)(1 − aq)(1 − aq2)...(1 − aqn−1), n ∈ N. Recently, the authors [7] defined the linear operator M(ai, bj ; q)f. Using the operatör M(ai, bj ; q)f(z)f, Aldweby and Darus [13] gave a new integral operator. In this work we highlight a result related to the new integral operator.