  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.
