Congruences for sequences analogous to Euler numbers

TL;DR

This paper introduces a generalized sequence extending Euler numbers, derives identities and inversion formulas, and establishes new congruences modulo powers of 2, 3, and 5 for the sequence's even terms.

Contribution

It generalizes Euler numbers through a new sequence, providing identities, inversion formulas, and novel congruences for its even-indexed terms.

Findings

Derived identities and inversion formulas for the sequence {E_{n,a}}.

Established congruences for E_{2n,a} modulo powers of 2, 3, and 5.

Extended properties of Euler numbers to a broader class of sequences.

Abstract

For a given real number $a$ we define the sequence $\{E_{n,a}\}$ by $E_{0,a}=1$ and $E_{n,a}=-a\sum_{k=1}^{[n/2]} \binom n{2k}E_{n-2k,a}$ $(n\ge 1)$, 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 $E_{2n,a}\mod{2^{{\rm ord}_2n+8}}$, $E_{2n,a}\pmod{3^{{\rm ord}_3n+5}}$ and $E_{2n,a}\pmod{5^{{\rm ord}_5n+4}}$ provided that $a$ is a nonzero integer, where ${\rm ord}_pn$ is the least nonnegative integer $\alpha$ such that $p^{\a}\mid n$ but $p^{\a+1}\nmid n$.