Extensions of Stern's congruence for Euler numbers

Abstract

For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\sum_{k=0}^{[n/2]}\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$ can be viewed as a generalization of Euler numbers. Let $k$ and $m$ be positive integers, and let $b$ be a nonnegative integer. In this paper, we determine $E_{2^mk+b}^{(a)}$ modulo $ 2^{m+10}$ for $m\ge 5$. For $m\ge 5$ we also establish congruences for $U_{k\varphi{(5^m)}+b},\; E_{k\varphi{(5^m)}+b},\; S_{k\varphi{(5^m)}+b}\pmod{5^{m+5}}$ and $S_{k\varphi{(3^m)}+b}\pmod{3^{m+5}},$ where $U_{2n}=E_{2n}^{(3/2)}$, $S_n=E_n^{(2)}$ and $\varphi(n)$ is Euler's function.