Abstract
We establish [.sub.r][PHI][.sub.s] as a general operator for many q-operators. A new polynomials [h.sub.n]([a.sub.1], * * *,[a.sub.r];[b.sub.1], * * *, [b.sub.s];x, y;q) are described as an extension of the bivariate Rogers-Szego polynomial [h.sub.n](x, y|q) and the generalized Al-Salam--Carlitz q-polynomials [PHI]n (x, y|q). With the use of the operator [.sub.r][PHI][.sub.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 [h.sub.n]([a.sub.1], * * *, [a.sub.r]; [b.sub.1], * * *,[b.sub.s];x, y; q). The generating function and its extension, Mehler's formula and its extension and Rogers formula and its extension for [h.sub.n](x, y|q) and [Please download the PDF to view the mathematical expression] are deduced by giving special values to parameters of a new polynomial [h.sub.n] ([a.sub.1], * * *, [a.sub.r]; [b.sub.1], * * *, [b.sub.s];x, y|q). Keywords: the q-operators, the bivariate Rogers-Szego polynomials, the generalized AlSalam--Carlitz q-polynomials, generating function, Mehler's formula, Rogers formula. AMS Subject Classification: 05A30, 33D45.