Parameter Augmentation for Basic Hypergeometric Series, II

Bivariate generating functions for Rogers–Szegö polynomials

“…and further generalized the corresponding results in [8,9]. For more information, please refer to [11,12].…”

“…Chen and Liu [8,9] introduced h = g À1 D q and the q-exponential operators EðbhÞ ¼ X 1 n¼0 q ð n 2 Þ ðq; qÞ n ðbh q Þ n ; TðbD q Þ ¼ X 1 n¼0 1 ðq; qÞ n ðbD q Þ n ; ð1:7Þ then they obtained many useful operator identities, please refer to [8,9,16]. In [21,22], the authors deduced that:…”

The Cauchy operator for basic hypergeometric series

“…We are ready to describe how one can employ the Cauchy operator to derive Mehler's formula and the Rogers formula for h n (x, y|q). [11,18,24,26] for the Rogers-Szegö polynomials.…”

“…Notice that when a = 0, the 2 φ 1 series on the right-hand side of (2.4) can be summed by employing the Cauchy q-binomial theorem (1.3). In this case (2.4) reduces to T (bD q ) 1 (cs, ct; q) ∞ = (bcst; q) ∞ (bs, bt, cs, ct; q) ∞ , |bs|, |bt| < 1, (2.5) which was derived by Chen and Liu in [11].…”

“…Recall that Chen and Liu [11] introduced the augmentation operator T (bD q ) = ∞ n=0 (bD q ) n (q; q) n (1.1) as the basis of parameter augmentation which serves as a method for proving q-summation and integral formulas from special cases for which some parameters are set to zero.…”