An Operator Approach to the Al-Salam-Carlitz Polynomials

TL;DR

This paper introduces an operator-based method to derive Rogers-type and Mehler's formulas for Al-Salam-Carlitz polynomials, simplifying derivations and extending known identities using q-operator techniques.

Contribution

It develops a novel operator approach employing q-exponential and augmentation operators to derive key formulas and identities for Al-Salam-Carlitz polynomials.

Findings

Derived Rogers-type formula using q-exponential operator.

Obtained Mehler's formula without the need for a terminating condition.

Extended identities for generating functions of Al-Salam-Carlitz polynomials.

Abstract

We present an operator approach to Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials $U_n(x,y,a;q)$. By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to by Al-Salam and Carlitz, which requires a terminating condition on a ${}_3\phi_2$ series. By means of the Cauchy companion augmentation operator, we obtain Mehler's formula in a similar form, but it does not need the terminating condition. We also give several identities on the generating functions for products of the Al-Salam-Carlitz polynomials which are extensions of formulas for Rogers-Szeg\"o polynomials.