## THE HOMOGENEOUS q-DIFFERENCE OPERATOR AND THE RELATED POLYNOMIALS

## THE HOMOGENEOUS q-DIFFERENCE OPERATOR AND THE RELATED POLYNOMIALS

*M. A. Arif, H. L. Saad*

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## Abstract

We create the homogeneous q-di erence operator e E(a; b; ) as an extension of the exponential operator E(b ). A new polynomials hn(a; b; xjq????1) are de ned as an extension of the q????1-Rogers-Szego polynomial hn(a; bjq????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; bjq????1). The generating function and its extension, Rogers formula and the invers linearization formula, and Mehler's formula for the polynomials hn(a; bjq????1) are deduced by giving special values to parameters of a new polynomial hn(a; b; xjq????1).

## Keywords

the homogeneous q-di erence operator, the q????1-Rogers-Szego polynomial, the generating function, the Rogers formula, the invers linearization formula, the Mehler's formula.