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

Z. K. Silagadze

doi:10.1515/gmj-2012-0023

paper

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

arXiv:1207.2055 · doi:10.1515/gmj-2012-0023

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 $ζ (2 n + 1)$ .

11 pages, to be published in Georgian Mathematical Journal. arXiv:1003.3602 is more extended version of this short note

Comment on the sums S(n)=k=−∞∑∞​(4k+1)n1​

Abstract

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