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

Author

Listed:
  • Jian Cao

    (School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China)

  • Jin-Yan Huang

    (School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China)

  • Mohammed Fadel

    (Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India)

  • Sama Arjika

    (Department of Mathematics and Informatics, University of Agadez, Agadez 900288, Niger)

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.

Suggested Citation

  • Jian Cao & Jin-Yan Huang & Mohammed Fadel & Sama Arjika, 2023. "A Review of q -Difference Equations for Al-Salam–Carlitz Polynomials and Applications to U ( n + 1) Type Generating Functions and Ramanujan’s Integrals," Mathematics, MDPI, vol. 11(7), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1655-:d:1111045
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    References listed on IDEAS

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    1. Jian-Ping Fang, 2014. "Remarks on Homogeneous Al-Salam and Carlitz Polynomials," Journal of Mathematics, Hindawi, vol. 2014, pages 1-12, July.
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    Cited by:

    1. Jung-Yoog Kang & Cheon-Seoung Ryoo, 2023. "Approximate Roots and Properties of Differential Equations for Degenerate q -Special Polynomials," Mathematics, MDPI, vol. 11(13), pages 1-14, June.

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