Abstract
In this paper we compute the values of series of the form\(S_k = \sum\limits_{n = 0}^\infty {\frac{{(2n + 1)^k }}{{1 + e^{(2n + 1)\pi } }}} \) if k ε N is add. This was done by Glaisher [4] if k≡1 (mod 4), but if k ≡3 (mod 4) the result seems to be new.
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Literatur
Tom M. Apostol: Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, Heidelberg, Berlin, 1976
Bruce C. Berndt: Eisenstein Series, Theta-functions and Series Relations in the Spirit of Ramanujan. Journal f.d. reine u. angew. Math., 303/304, 1978
G.V. Cudnovskii: Siehe etwa Seminaire Bourbaki, Exposé n∘ 488, (28 année, juin 1976)
J.W.I. Glaisher: On the Series which represent the twelve elliptic and four zeta functions, Mess. Math.18, 1889.
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Schoißengeier, J. Der numerische Wert gewisser Reihen. Manuscripta Math 38, 257–263 (1982). https://doi.org/10.1007/BF01168594
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DOI: https://doi.org/10.1007/BF01168594