Comment on the sums $S(n)=\sum\limits_{k=-\infty}^\infty \frac{1}{(4k+1)^n}$

TL;DR

This paper simplifies the calculation of a specific infinite sum involving powers of integers and provides an explicit kernel expression that aids in deriving representations for odd zeta values.

Contribution

It offers an explicit kernel expression for an integral operator, simplifying the computation of the sum S(n) and deriving integral representations for ζ(2n+1).

Findings

Explicit kernel expression for the integral operator.

Simplified calculation of the sum S(n).

Derived integral representation for ζ(2n+1).

Abstract

This is a comment on the papers N. D. Elkies, Amer. Math. Monthly 110 (2003), 561-573 and Cvijovic and J. Klinowski, J. Comput. Appl. Math. 142 (2002), 435-439. We provide an explicit expression for the kernel of the integral operator introduced in the first paper. This explicit expression considerably simplifies the calculation of S(n) and enables a simple derivation of Cvijovic and Klinowski's integral representation for $\zeta(2n+1)$.