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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.

We introduce poly-Bernoulli polynomials in two variables by using a generalization of Stirling numbers of the second kind that we studied in a previous work. We prove the bi-variate poly-Bernoulli polynomial version of some known results on standard Bernoulli polynomials, as the addition formula and the binomial formula. We also prove a result that allows us to obtain poly-Bernoulli polynomial identities from polynomial identities, and we use this result to obtain several identities involving products of poly-Bernoulli and/or standard Bernoulli polynomials. We prove two generalized recurrences for bi-variate poly-Bernoulli polynomials, and obtain some corollaries from them.

We introduce poly-Cauchy numbers with level 2. Poly-Cauchy numbers may be interpreted as a kind of generalizations of the classical Cauchy numbers by using the inverse relation of exponentials and ...

We introduce and study shifted Cauchy numbers as different natural extension of the classical Cauchy numbers, in particular, in terms of determinantal expressions. We give several arithmetical or c...

We define a truncated Euler polynomial E m,n (x) as a generalization of the classical Euler polynomial En (x). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

We give elementary proofs of three formulas involving Bell numbers, including a generalization of the Gould-Quaintance formula and a generalization of Spivey?s formula. We find variants for two of our formulas which involve some well-known sequences, among them the Fibonacci, Bernoulli and Euler numbers.