A congruence concerning a convolution involving weighted Bernoulli numbers

TL;DR

This paper establishes a new congruence involving convolutions of weighted Bernoulli numbers modulo a prime p, using p-adic analysis of permutation counts with ascents.

Contribution

It introduces a novel congruence relating weighted Bernoulli numbers and harmonic numbers via p-adic expansion techniques.

Findings

Derived a congruence for convolutions of Bernoulli numbers modulo p

Connected permutation ascent counts to Bernoulli number properties

Utilized p-adic expansions to prove the main result

Abstract

Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the p-adic expansion of the number of permutations of Sym(p-2) with an even number of ascents, up to the modulus $p^2$.