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arXiv · 2201.10117

Generalized $q$-Bernoulli polynomials generated by Jackson $q$-Bessel functions

S. Z. Eweis·Zeinab S. I. Mansour
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Abstract

In this paper, we introduce the polynomials $B^{(k)}_{n,\alpha}(x;q)$ generated by a function including Jackson $q$-Bessel functions $J^{(k)}_{\alpha}(x;q)$ $ (k=1,2,3),\,\alpha>-1$. The cases $\alpha=\pm\frac{1}{2}$ are the $q$-analogs of Bernoulli and Euler$^{,}$s polynomials introduced by Ismail and Mansour for $(k=1,2)$, Mansour and Al-Towalib for $(k=3)$. We study the main properties of these polynomials, their large $n$ degree asymptotics and give their connection coefficients with the $q$-Laguerre polynomials and little $q$-Legendre polynomials.

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BibTeXRIS

S. Z. Eweis, Zeinab S. I. Mansour. 2022-01-25. Generalized $q$-Bernoulli polynomials generated by Jackson $q$-Bessel functions. https://arxiv.org/abs/2201.10117

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