The Operator S(a,b;θ_x ) for the Polynomials  Z_n (x,y,a,b;q)

Husam L. Saad; Faiz A. Reshem

doi:10.24996/ijs.2022.63.10.26

Authors

Husam L. Saad Department of Mathematics, College of Science, Basrah University, Basrah, Iraq https://orcid.org/0000-0001-8923-4759
Faiz A. Reshem Department of Mathematics, College of Science, Basrah University, Basrah, Iraq

DOI:

https://doi.org/10.24996/ijs.2022.63.10.26

Keywords:

The bivariate Rogers-Szegö polynomials, Generating function, Rogers formula, Inverse linearisation formula, q-difference equation

Abstract

In this work, we give an identity that leads to establishing the operator . Also, we introduce the polynomials . In addition, we provide Operator proof for the generating function with its extension and the Rogers formula for . The generating function with its extension and the Rogers formula for the bivariate Rogers-Szegö polynomials are deduced. The Rogers formula for allows to obtain the inverse linearization formula for , which allows to deduce the inverse linearization formula for . A solution to a q-difference equation is introduced and the solution is expressed in terms of the operators . The q-difference method is used to recover an identity of the operator and the generating function for the polynomials .