APPLICATION OF THE OPERATOR φ (a,b,c d,e ; q, f Dq) FOR THE POLYNOMIALS Yn(a, b, c; d, e; x, y|q)

In this paper, we construct the exponential operator phi ((a,b,c)( d,e) ; q, f D-q) d that has five parameters a, b, c, d, e and we define a more general polynomials Y-n (a, b, c; d, e; x, y vertical bar q), in which case, the bivariate Rogers-Szego polynomials h(n) (x, y vertical bar q) become special cases of Y-n (a, b, c; d, e; x, y vertical bar 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 Y-n (a, b, c; d, e; x, y vertical bar 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 Y-n (a, b, c; d, e; x, y vertical bar q) in order to establish the generating function and its extension, Mehler's formula and its extension, and the Rogers formula for h(n)(x, y vertical bar q).