Congruences for sequences analogous to Euler numbers

Abstract

For a given real number $a$ we define the sequence $\{E_{n,a}\}$ by $E_{0,a}=1$ and $E_{n,a}=-a{\sum }_{k=1}^{[n/2]}\left(\genfrac{}{}{0ex}{}{n}{2k}\right)E_{n-2k,a}$ $(n\ge 1)$, where $[x]$ is the greatest integer not exceeding $x$. Since $E_{n,1}=E_n$ is the n-th Euler number, $E_{n,a}$ can be viewed as a natural generalization of Euler numbers. In this paper we deduce some identities and an inversion formula involving $\{E_{n,a}\}$, and establish congruences for $E_{2n,a}\mathrm{mod}{2}^{{\mathrm{ord}}_{2}n+8}$, $E_{2n,a}(\mathrm{mod}{3}^{{\mathrm{ord}}_{3}n+5})$ and $E_{2n,a}(\mathrm{mod}{5}^{{\mathrm{ord}}_{5}n+4})$ provided that $a$ is a nonzero integer, where ${\mathrm{ord}}_{p}n$ is the least nonnegative integer $\alpha$ such that $p^{\a}\mid n$ but $p^{\a+1}\nmid n$.

16 pages