A note on generalized $q$-difference equations for general Al-Salam--Carlitz polynomials

TL;DR

This paper derives generalized $q$-difference equations for Al-Salam--Carlitz polynomials, extends recent results, and establishes new identities, generating functions, and integrals involving these polynomials using $q$-difference methods.

Contribution

It introduces generalized $q$-difference equations for Al-Salam--Carlitz polynomials and extends recent work, providing new identities and generating functions.

Findings

Derived generalized $q$-difference equations for the polynomials.

Obtained transformational identities using $q$-difference equations.

Established $U(n+1)$ type generating functions and Ramanujan's integrals.

Abstract

In this paper, we deduce the generalized $q$-difference equations for general Al-Salam--Carlitz polynomials and generalize Arjika's recently results [$q$-difference equation for homogeneous $q$-difference operators and their applications, J. Differ. Equ. Appl. {\bf 26}, 987--999 (2020)]. In addition, we obtain transformational identities by the method of $q$-difference equation. Moreover, we deduce $U(n+1)$ type generating functions and Ramanujan's integrals involving general Al-Salam--Carlitz polynomials by $q$-difference equation.