Applications of the operator r?s in q-polynomials

Abstract

We establish r?s as a general operator for many q-operators. A new polynomials hn(a1, · · · , ar; b1, · · · , bs; x, y; q) are described as an extension of the bivariate Rogers-Szegö polynomial hn(x, y|q) and the generalized Al-Salam–Carlitz q-polynomials ? n (a,b) (x, y|q). With the use of the operator r?s, we provide an operator proof of the generating function and its extension, Mehler’s formula and its extension and Rogers formula and its extension to the polynomials hn(a1, · · · , ar; b1, · · · , bs; x, y; q). The generating function and its extension, Mehler’s formula and its extension and Rogers formula and its extension for hn(x, y|q) and ? n (a,b) (x, y|q) are deduced by giving special values to parameters of a new polynomial hn(a1, · · · , ar; b1, · · · , bs; x, y|q).