Generalized q-difference equations for (q, c)-hypergeometric polynomials and some applications - Cao, Jian; Zhou, Hong-Li; Arjika, Sama (The Ramanujan journal, 2023)
In this paper, our investigation is motivated by the concept of ( q , c )-derivative operators introduced by Zhang (Adv Appl Math 121:102081, 2020). Then we seek and find that ( q , c )-hypergeometric polynomials involving ( q , c )-derivative operators are solutions of certain generalized q -difference equations. We introduce two homogeneous ( q , c )-difference operators T c ( a , b , d , u , v , x D c , y ) and E c ( a , b , d , u , v , x θ c , y ) , which turn out to be suitable for studying two families of generalized ( q , c )-Al-Salam–Carlitz polynomials Φ n ( a , b , d , u , v , c ) ( x , y | q ) and Υ n ( a , b , d , u , v , c ) ( x , y | q ) . Several q -identities such as: generating functions, Andrews–Askey integrals and U ( n + 1 ) type q -binomial formulas for generalized q -polynomials are derived by the method of ( q , c )-difference equations.