A Review of q-Difference Equations for Al-Salam-Carlitz Polynomials and Applications to U(n+1) Type Generating Functions and Ramanujan's Integrals

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-Szego 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.

Automated summary

In this review paper, we study the progress and extensions of q-difference equations for generalized Al-Salam-Carlitz polynomials related to theta functions, q-exponential operators, and Rogers-Szego polynomials. We provide proofs, transformational identities, generating functions, and integrals involving these polynomials using q-difference equations. Furthermore, we derive extensions of the Andrews-Askey integral through q-difference equations.