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.
Categories: math.NT