The q-Exponential Operator and Generalized Rogers-Szego Polynomials - Saad, Husam L. (JOURNAL OF ADVANCES IN MATHEMATICS, 2014)
This paper is mainly concerned with using q-exponential operator T(bDq) in proving the identities that involve the generalized Rogers-Szego polynomials rn(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 rn(x,b)of given in [7, 8] which are: generating function, Mehler's formula and Rogers formula. Then we introduce several extensions of rn(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 hn(x|q) when b=1.