Generalized $q$-Bernoulli polynomials generated by Jackson $q$-Bessel functions
In this paper, we introduce the polynomials $B^{(k)}_{n,α}(x;q)$ generated by a function including Jackson $q$-Bessel functions $J^{(k)}_α(x;q)$ $ (k=1,2,3),\,α>-1$. The cases $α=\pm\frac{1}{2}$ 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.
math.CA↗