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

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.

14 pages

References in corpus (5)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Some homogeneous $q$-difference operators and the associated generalized Hahn polynomials
• A Note on Generalized $q$-Difference Equations and Their Applications Involving $q$-Hypergeometric Functions
• Generating Functions for Some Families of the Generalized Al-Salam-Carlitz $q$-Polynomials
• $q$-difference equations for homogeneous $q$-difference operators and their applications

Cited by in corpus (2)

• A Note on Generalized $q$-Difference Equations and Their Applications Involving $q$-Hypergeometric Functions
• A General Family of $q$-Hypergeometric Polynomials and Associated Generating Functions