On convolutions of Euler numbers

Abstract

We show that if p is an odd prime then $$\sum _{k=0}^{p-1}{E}_k{E}_{p-1-k}=1(modp)$$ and $$\sum _{k=0}^{p-3}{E}_k{E}_{p-3-k}=(-1{)}^{(p-1)/2}2E_{p-3}(modp),$$ 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 $$\sum _{k=0}^{p-1+2n}{E}_k{E}_{p-1+2n-k}=s(n)(modp)$$ where s(n) is an integer only depending on n.

6 pages