Applications of the generalized homogeneous q-shift operator in q-polynomials

Abstract

In this paper, we construct the generalized homogeneous q-shift operatör r?s ( a1, · · · , ar b1, · · · , bs ; q, cDxy). Then, we apply this operator to derive some q-identities such as: the generating function and its extension, Rogers formula and its extension, Mehler’s formula and its extension, Srivastava-Agarwal type bilinear generating functions for the polynomials ?(a,b)n (x, y, c|q). Also, we obtain a transformation formula involving generating functions for ?(a,b)n (x, y, c|q). We provide some special values for ?(a,b)n (x, y, c|q) in order to establish identities for the polynomials ?(a)n (x) and ?(a,b)n (x, y|q).