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An expansion of \((q, \lambda )\)-derivative operator

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Abstract

We give the concept of Generalized Rogers–Szegö polynomials based on the \((q, \lambda )\)-derivative operator and \((q, \mu )\)-derivative operator. Then we use the method of Liu’s calculus to obtain the expansion theorem involving Generalized Rogers–Szegö polynomials. In addition, we use two kinds of the \((q, \lambda )\)-exponential functions and extend some identities of Zhang. At last, we consider the properties when \(\lambda =1\) and \(\mu =1\), and use the method of the generating functions and q-Mehler formula for Al-Salam–Carlitz polynomials to establish some new identities.

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Acknowledgements

I would also like to thank Professor Zhiguo Liu for technical and material support during the writing of the manuscript and thank my schoolmate Shiyang Weng for Corollary 5.1 and Corollary 5.2. The author also thanks the reviewer and my schoolmate Deliang Wei for many suggestions for the revision of this paper.

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  1. Department of Mathematics, East China Normal University, 500 Dongchuan Roda, Shanghai, 200241, People’s Republic of China

    Dunkun Yang

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  1. Dunkun Yang
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Yang, D. An expansion of \((q, \lambda )\)-derivative operator. Ramanujan J 60, 1127–1149 (2023). https://doi.org/10.1007/s11139-022-00617-w

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