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Generalized q-difference equations for (q, c)-hypergeometric polynomials and some applications

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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 \({\mathbb {T}}_c(a,b,d,u,v,xD_{c,y})\) and \({\mathbb {E}}_c(a,b,d,u,v,x\theta _{c,y})\), which turn out to be suitable for studying two families of generalized (q, c)-Al-Salam–Carlitz polynomials \(\Phi _n^{(a,b,d, u,v,c)}(x,y|q)\) and \(\Upsilon _n^{(a,b,d, u,v,c)}(x,y|q)\). 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.

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Acknowledgements

The author would like to express deep appreciation to the referee and the editor for their helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, Zhejiang, People’s Republic of China

    Jian Cao & Hong-Li Zhou

  2. Department of Mathematics and Informatics, University of Agadez, Post Office Box 199, Agadez, Niger

    Sama Arjika

  3. International Chair of Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Post Box 072, Cotonou 50, Godomey, Benin

    Sama Arjika

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  1. Jian Cao
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  2. Hong-Li Zhou
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  3. Sama Arjika
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Correspondence to Jian Cao.

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This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY21A010019) and the National Natural Science Foundation of China (Grant No. 12071421)

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Cao, J., Zhou, HL. & Arjika, S. Generalized q-difference equations for (q, c)-hypergeometric polynomials and some applications. Ramanujan J 60, 1033–1067 (2023). https://doi.org/10.1007/s11139-022-00634-9

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