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–Szegö polynom...

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Bibliographic Details
Published inMathematics (Basel) Vol. 11; no. 7; p. 1655
Main Authors Cao, Jian, Huang, Jin-Yan, Fadel, Mohammed, Arjika, Sama
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.04.2023
Subjects
Al-Salam–Carlitz polynomials
Difference equations
Functions (mathematics)
generating functions
Identities
Integrals
Mathematical analysis
Operators (mathematics)
Polynomials
q-difference equation
q-exponential operator
Ramanujan’s integral
Variables
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math11071655