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Two q-Operational Equations and Hahn Polynomials

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Abstract

Motivated by Liu’s (Sci China Math 66:1199–1216, 2023) recent work. This article reveals the essential features of Hahn polynomials by presenting a new q-exponential operator, that is

$$\begin{aligned} \exp _q(t\Delta _{x,a})f(x)=\frac{(axt;q)_{\infty }}{(xt;q)_{\infty }} \sum _{n=0}^{\infty }\frac{t^n}{(q;q)_n} f(q^n x) \end{aligned}$$

with \(\Delta _{x,a}=x (1-a)\eta _a+\eta _x\) and \(\eta _x \{f(x) \}=f(qx)\). Letting \(f(x) \equiv 1\) and the above operator equation immediately becomes the generating function of Hahn polynomials. These lead us to use a systematic method for studying identities involving Hahn polynomials. As applications, we use the method of the q-exponential operator to prove some new q-identities, including q-Nielsen’s formulas and Carlitz’s extension for the Hahn polynomials, etc. Moreover, a generalization of q-Gauss summation is given, too.

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Acknowledgements

The authors are grateful to the anonymous referee for helpful suggestions to improve the paper. They also sincerely thank Professor ZhiGuo Liu for his encouragement and valuable suggestions.

Funding

This work was supported by the National Science Foundation of P. R. China (Grant No. 12371328).

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

    Jing Gu, DunKun Yang & Qi Bao

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  1. Jing Gu
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Contributions

Yang and Bao proposed Theorem 1.2 as the beginning of the manuscript. Gu mainly calculates the details of the manuscript. Yang gives us some ideas for some complicated calculations such as Theorem 3.3. Bao finally compiled and completed the manuscript. Everyone has checked the details of the manuscript.

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Correspondence to Qi Bao.

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Communicated by Uwe Kaehler.

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Gu, J., Yang, D. & Bao, Q. Two q-Operational Equations and Hahn Polynomials. Complex Anal. Oper. Theory 18, 53 (2024). https://doi.org/10.1007/s11785-024-01496-3

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