A Comment on Matiyasevich's Identity #0102 with Bernoulli Numbers

TL;DR

This paper generalizes Matiyasevich's identity involving Bernoulli numbers, connecting it with Ramanujan summation and deriving new recursion relations through advanced analytic techniques.

Contribution

It introduces a unified framework linking Bernoulli identities with Ramanujan summation, providing novel recursion formulas for Bernoulli number derivatives.

Findings

Generalized Matiyasevich's identity with Bernoulli numbers

Established new recursion relations for Bernoulli derivatives

Connected divergent series summation with Bernoulli identities

Abstract

We connect and generalize Matiyasevich's identity #0102 with Bernoulli numbers and an identity of Candelpergher, Coppo and Delabaere on Ramanujan summation of the divergent series of the infinite sum of the harmonic numbers. The formulae are analytic continuation of Euler sums and lead to new recursion relations for derivatives of Bernoulli numbers. The techniques used are contour integration, generating functions and divergent series.