Moments of the cot function, central factorial numbers and their links with the Dirichlet eta function at odd integers

TL;DR

This paper explores the moments of the cotangent function through central factorial numbers, introducing new integral representations that connect these moments to Dirichlet eta functions at odd integers.

Contribution

It presents a novel integral representation of central factorial numbers, linking cot moments to Dirichlet eta functions and enabling recursive computation methods.

Findings

Derived new integral representations for central factorial numbers

Expressed cot moments as linear combinations of Dirichlet eta functions

Enabled recursive computation of harmonic series and integrals

Abstract

We investigate the properties of the moments of the cot function using the central factorial numbers. Using a new integral representation of the central factorial numbers, we find a new way to express these moments in terms of recursive sums and integrals. This allows us to compute 'recursive' generalized harmonic series and multiple integrals as a linear combination of the Dirichlet eta functions at odd integers.