On convolutions of Euler numbers

TL;DR

This paper investigates modular identities involving Euler numbers and convolutions, establishing new congruences for sums of products of Euler numbers modulo primes, extending known results to broader cases.

Contribution

The paper introduces new congruences for convolutions of Euler numbers modulo primes, generalizing previous results and providing explicit formulas depending on the parameter n.

Findings

Proves specific sums of Euler number products are congruent to 1 or a multiple of Euler numbers modulo p.

Establishes a general formula for sums involving Euler numbers and primes greater than 2n+1.

Provides explicit congruences depending on the parity of (p-1)/2.

Abstract

We show that if p is an odd prime then $$\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer n and prime number p>2n+1 we have $$\sum_{k=0}^{p-1+2n}E_kE_{p-1+2n-k}=s(n) (mod p)$$ where s(n) is an integer only depending on n.