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

被引：10

作者：

Cao, Jian ^{[1
]}

Huang, Jin-Yan ^{[1
]}

Fadel, Mohammed ^{[2
]}

Arjika, Sama ^{[3
]}

机构：

[1] Hangzhou Normal Univ, Sch Math, Hangzhou 311121, Peoples R China

[2] Aligarh Muslim Univ, Dept Math, Aligarh 202002, India

[3] Univ Agadez, Dept Math & Informat, Agadez 900288, Niger

来源：

MATHEMATICS | 2023年 / 11卷 / 07期

基金：

中国国家自然科学基金;

关键词：

q-difference equation; q-exponential operator; Al-Salam-Carlitz polynomials; generating functions; Ramanujan's integral; OPERATOR IDENTITIES; TRANSFORMATION; EXPANSIONS; EXTENSION; SUMMATION; SERIES; BETA;

D O I：

10.3390/math11071655

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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.

引用

页数：22

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