The q-Exponential Operator and Generalized Rogers-Szegö Polynomials

Abstract

This paper is mainly concerned with using -exponential operator in proving the identities that involve the generalized Rogers-Szego polynomials . We introduce some new roles of the -exponential operator and prove that the generalized Rogers-Szego polynomials can be represented by the -exponential operator, so we use this operator and it’s roles in proving the basic identities of given in [7, 8] which are: generating function, Mehler’s formula and Rogers formula. Then we introduce several extensions of identities such that: the extended generating function, extended Mehler’s formula, extended Rogers formula and another extended identities. These extended identities of the generalized Rogers-Szego polynomials can be considered a general form of the corresponding identities for the classical Rogers-Szego polynomials when b=1.