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

Abstract

Abstract In this paper, we deduce the generalized q-difference equations for general Al-Salam–Carlitz polynomials and generalize Arjika’s recent results (Arjika in J. Differ. Equ. Appl. 26:987–999, 2020). In addition, we obtain transformational identities by the method of q-difference equation. Moreover, we deduce U ( n + 1 ) $U(n+1)$ type generating functions and Ramanujan’s integrals involving general Al-Salam–Carlitz polynomials by q-difference equation.

Keywords

• q-Difference equation
• q-Difference operator
• Al-Salam–Carlitz polynomials
• Generating functions
• Ramanujan’s integral