Generalized Kronecker formula for Bernoulli numbers and self-intersections of curves on a surface

TL;DR

This paper introduces a new explicit formula for Bernoulli numbers involving two parameters, generalizing the Kronecker formula, with proofs rooted in analysis and topology of surface curves.

Contribution

It provides a novel generalized formula for Bernoulli numbers and offers two distinct proofs, one analytic and one algebraic, inspired by topology.

Findings

New explicit formula for Bernoulli numbers with parameters

Reduction to Kronecker formula when parameters are specific

Two different proofs: analytic and topological

Abstract

We present a new explicit formula for the $m$-th Bernoulli number $B_m$, which involves two integer parameters $a$ and $n$ with $0\le a\le m\le n$. If we set $a=0$ and $n=m$, then the formula reduces to the celebrated Kronecker formula for $B_m$. We give two proofs of our formula. One is analytic and uses a certain function in two variables. The other is algebraic and is motivated by a topological consideration of self-intersections of curves on an oriented surface.