## APPLICATION OF THE OPERATOR ∅(█(a,b,c@d,e);q,〖f D〗_q ) FOR THE POLYNOMIALS Y_n(a, b, c; d, e; x, ├ y┤|q)

## APPLICATION OF THE OPERATOR ∅((a,b,c@d,e);q,〖f D〗_q ) FOR THE POLYNOMIALS Y_n(a, b, c; d, e; x, ├ y┤|q)

*H. L. Saad, R. H. Jaber*

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## Abstract

In this paper, we construct the exponential operator a; b; c d; e ; q; fDq that has ve parameters a, b, c, d, e and we de ne a more general polynomials Yn(a; b; c; d; e; x; yjq), in which case, the bivariate Rogers-Szego polynomials hn(x; yjq) become special cases of Yn(a; b; c; d; e; x; yjq). Furthermore, we involve the operator's technique to give an elegant proof for the generating function with its extension, Mehler's formula with its extension, and Rogers formula for the polynomials Yn(a; b; c; d; e; x; yjq). As well as, we present some special values for the parameters a, b, c, d, e that will be inserted in the identities of Yn(a; b; c; d; e; x; yjq) in order to establish the generating function and its extension, Mehler's formula and its extension, and the Rogers formula for hn(x; yjq).

## Keywords

The bivariate Rogers-Szego polynomials, the generating function, Mehler's formula, Rogers formula.