<i>q</i>-Difference equations of moment integrals for transformational identities and generating functions

Cao, Jian; Cai, Tianxin

doi:10.1080/10236198.2014.961444

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2023

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“…This theory is called Liu's calculus by Aslan and Ismail [20]. For further information about q-difference equations and q-partial difference equations, see the details in [10,17,[19][20][21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

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

2023

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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.

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