The q-Exponential Operator and Generalized Rogers-Szego Polynomials

Abstract

This paper is mainly concerned with using q-exponential operator T(bD_q) in proving the identities that involve the generalized Rogers-Szego polynomials  r_n(x,b) . We introduce some new roles of the q -exponential operator and prove that the generalized Rogers-Szego polynomials can be represented by theq -exponential operator, so we use this operator and it's roles in proving the basic identities r_n(x,b)of  given in [7, 8] which are: generating function, Mehler's formula and Rogers formula. Then we introduce several extensions  of r_n(x,b)  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 h_n(x|q)  when b=1.