Several formulas for Bernoulli numbers and polynomials

A generalized Stirling numbers of the second kind \begin{document}$ S_{a,b}\left(p,k\right) $\end{document} , involved in the expansion \begin{document}$ \left(an+b\right)^{p} = \sum_{k = 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} $\end{document} , where \begin{document}$ a \neq 0, b $\end{document} are complex numbers, have studied in [ 16 ]. In this paper, we show that Bernoulli polynomials \begin{document}$ B_{p}(x) $\end{document} can be written in terms of the numbers \begin{document}$ S_{1,x}\left(p,k\right) $\end{document} , and then use the known results for \begin{document}$ S_{1,x}\left(p,k\right) $\end{document} to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.