A note on the higher derivatives of the function 1/(exp(x) - 1)

TL;DR

This paper compares two formulas for higher derivatives of 1/(exp(x)-1), provides an integral representation, and links zeta values with Bernoulli numbers, enhancing understanding of special functions.

Contribution

It introduces a new integral representation and compares existing formulas for derivatives, connecting special functions with classical number theory.

Findings

Derived an integral representation for derivatives

Compared two existing formulas for higher derivatives

Established a relation between zeta values and Bernoulli numbers

Abstract

In this note we compare two formulas for the higher order derivatives of the function 1/(exp(x) -1). We also provide an integral representation for these derivatives and obtain a classical formula relating zeta values and Bernoulli numbers.