## APPLICATIONS OF THE OPERATOR r????s IN q-POLYNOMIALS

## APPLICATIONS OF THE OPERATOR r????s IN q-POLYNOMIALS

*H. L. Saad, H. J. Hassan*

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## 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-Szego polynomial hn(x; yjq) and the generalized Al-Salam{Carlitz q-polynomials (a;b) n (x; yjq). 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; yjq) and (a;b) n (x; yjq) are deduced by giving special values to parameters of a new polynomial hn(a1; ; ar; b1; ; bs; x; yjq).

## Keywords

the q-operators, the bivariate Rogers-Szego polynomials, the generalized Al- Salam{Carlitz q-polynomials, generating function, Mehler's formula, Rogers formula.