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Generalized $q$-Bernoulli polynomials generated by Jackson $q$-Bessel functions (2201.10117v1)
Published 25 Jan 2022 in math.CA, math-ph, math.FA, and math.MP
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.