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

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.

On q-Hermite Polynomials with Three Variables: Recurrence Relations, q-Differential Equations, Summation and Operational Formulas

“…Comparing the identical powers of t on each side gives rise to statement (79). Using Equation (17) once more, we obtain E q (−yt 2 ) e q (xt)e q (yt 2 )e q (zt 3 ) = e q (xt)e q (zt 3 ).…”

“…The debut of q-calculus allows for the emergence and investigation of the q-analogues that represent different elementary and special functions. Recently, several scientists have examined and studied certain special polynomials associated with q-calculus [16][17][18][19][20][21][22][23][24].…”

A note on $q$-truncated exponential polynomials

“…The q-analogue of several special functions like q-Hermite polynomials, q-Laguerre polynomials, q-Appell polynomials and q-Sheffer polynomials are established and studied. Very recently, the quantum algebra representations of certain q-special functions like q-Tricomi functions, 2-variable q-Bessel functions, 2-variable q-Hermite polynomials, 2-variable q-Laguerre polynomials, family of q-modified-Laguerre-Appell polynomials, characterizing q-Bessel functions of the first kind and a review on q-difference equations for Al-Salam-Carlitz polynomials are obtained [4,5,11,12,[24][25][26].…”

Boundary Value Problem for a Coupled System of Nonlinear Fractional q-Difference Equations with Caputo Fractional Derivatives

“…For detailed discussions and examples of nonlinear fractional q-difference equations subject to various boundary conditions involving q-derivatives and q-integrals, the book by Annaby and Mansour [12] is a valuable resource. Furthermore, extensive research has been conducted on q-difference and fractional q-difference equations, as evidenced by works such as [13][14][15].…”