Application of the operator φ ((a,b,c@d,e); q,ƒDq) for the polynomials Yn(a, b, c; d, e; x, y|q)

Abstract

In this paper, we construct the exponential operator φ((a,b,c@d,e); q,ƒDq) that has five parameters a, b, c, d, e and we define a more general polynomials Yn(a, b, c; d, e; x, y|q), in which case, the bivariate Rogers-Szeg¨o polynomials hn(x, y|q) become special cases of Yn(a, b, c; d, e; x, y|q). 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, y|q). 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, y|q) in order to establish the generating function and its extension, Mehler’s formula and its extension, and the Rogers formula for hn(x, y|q).