Extensions of Stern's congruence for Euler numbers

TL;DR

This paper extends Stern's congruence for Euler numbers by deriving new modular relations for generalized Euler numbers and related sequences, providing explicit congruences modulo high powers of 2, 3, and 5.

Contribution

It introduces new congruences for generalized Euler numbers and related sequences, extending Stern's congruence to higher powers of primes and specific indices.

Findings

Determines $E_{2^mk+b}^{(a)}$ modulo $2^{m+10}$ for $m extgreater=5$.

Establishes congruences for sequences $U_{k extphi(5^m)+b}$, $E_{k extphi(5^m)+b}$, and $S_{k extphi(5^m)+b}$ modulo $5^{m+5}$ and $3^{m+5}$.

Provides explicit formulas for generalized Euler numbers at specific indices.

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.