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

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 $\zeta(2n+1)$.