Comment on the sums $S(n)=\sum\limits_{k=-\infty}^\infty \frac{1}{(4k+1)^n}$
Classical Analysis and ODEs
2012-09-11 v1
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 .
Cite
@article{arxiv.1207.2055,
title = {Comment on the sums $S(n)=\sum\limits_{k=-\infty}^\infty \frac{1}{(4k+1)^n}$},
author = {Z. K. Silagadze},
journal= {arXiv preprint arXiv:1207.2055},
year = {2012}
}
Comments
11 pages, to be published in Georgian Mathematical Journal. arXiv:1003.3602 is more extended version of this short note