A Review of <i>q</i>-Difference Equations for Al-Salam–Carlitz Polynomials and Applications to <i>U</i>(<i>n</i> + 1) Type Generating Functions and Ramanujan’s Integrals

Abstract

In this review paper, our aim is to study the current research progress of q-difference equations for generalized Al-Salam–Carlitz polynomials related to theta functions and to give an extension of q-difference equations for q-exponential operators and q-difference equations for Rogers–Szegö polynomials. Then, we continue to generalize certain generating functions for Al-Salam–Carlitz polynomials via q-difference equations. We provide a proof of Rogers formula for general Al-Salam–Carlitz polynomials and obtain transformational identities using q-difference equations. In addition, we gain U(n+1)-type generating functions and Ramanujan’s integrals involving general Al-Salam–Carlitz polynomials via q-difference equations. Finally, we derive two extensions of the Andrews–Askey integral via q-difference equations.

Keywords

• <i>q</i>-difference equation
• <i>q</i>-exponential operator
• Al-Salam–Carlitz polynomials
• generating functions
• Ramanujan’s integral