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

Jian Cao (), Jin-Yan Huang, Mohammed Fadel and Sama Arjika
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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

Mathematics, 2023, vol. 11, issue 7, 1-22

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: q -difference equation; q -exponential operator; Al-Salam–Carlitz polynomials; generating functions; Ramanujan’s integral (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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