Two operator representations for the trivariate q-polynomials and Hahn polynomials

Abstract

In this paper, we introduce a trivariate q-polynomials $F_n(x,y,z;q)$ as a general form of Hahn polynomials $\psi _n^{(a)}(x|q)$ and $\psi _n^{(a)}(x,y|q)$. We represent $F_n(x,y,z;q)$ by two operators: the homogeneous q-shift operator $L(b\theta _{xy})$ given by Saad and Sukhi (Appl Math Comput 215:4332–4339, 2010), and the Cauchy companion operator $E(a,b;\theta )$ given by Chen (q-Difference Operator and Basic Hypergeometric Series, 2009) to derive the generating function, symmetric property, Mehler’s formula, Rogers formula, another Roger-type formula, linearization formula, and an extended Rogers formula for the trivariate q-polynomials. Then, we give the corresponding formulas for our new definitions of Hahn polynomials $\psi _n^{(a)}(x|q)$ and $\psi _n^{(a)}(x,y|q)$ by representing Hahn polynomials by the operators $L(b\theta _{xy})$ and $E(a,b;\theta )$, and by a special substitution in the trivariate q-polynomials $F_n(x,y,z;q)$.