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A q-operational equation and the Rogers-Szegő polynomials

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Abstract

By solving a q-operational equation with formal power series, we prove a new q-exponential operational identity. This operational identity reveals an essential feature of the Rogers-Szegő polynomials and enables us to develop a systematic method to prove the identities involving the Rogers-Szegő polynomials. With this operational identity, we can easily derive, among others, the q-Mehler formula, the q-Burchnall formula, the q-Nielsen formula, the q-Doetsch formula, the q-Weisner formula, and the Carlitz formula for the Rogers-Szegő polynomials. This operational identity also provides a new viewpoint on some other basic q-formulas. It allows us to give new proofs of the q-Gauss summation and the second and third transformation formulas of Heine and give an extension of the q-Gauss summation.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11971173) and Science and Technology Commission of Shanghai Municipality (Grant No. 13dz2260400). The author is grateful to the referees for many very helpful comments and suggestions.

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  1. School of Mathematical Sciences, East China Normal University, Shanghai, 200241, China

    Zhiguo Liu

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  1. Zhiguo Liu
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Liu, Z. A q-operational equation and the Rogers-Szegő polynomials. Sci. China Math. 66, 1199–1216 (2023). https://doi.org/10.1007/s11425-021-1999-2

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  • DOI: https://doi.org/10.1007/s11425-021-1999-2

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