The homogeneous q-difference operator and the related polynomials

Abstract

We create the homogeneous q-difference operator Ee(a, b; ?) as an extension of the exponential operator E(b?). A new polynomials hn(a, b, x|q?1) are defined as an extension of the q?1-Rogers-Szegö polynomial hn(a, b|q?1). We provide an operator proof of the generating function and its extension, Rogers formula and the invers linearization formula, and Mehler’s formula for the polynomials hn(a, b|q?1). The generating function and its extension, Rogers formula and the invers linearization formula, and Mehler’s formula for the polynomials hn(a, b|q?1) are deduced by giving special values to parameters of a new polynomial hn(a, b, x|q?1).