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The structural elucidation of Eisenstein’s formula

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Abstract

In this paper, we shall give a complete structural description of generalizations of the classical Eisenstein formula that expresses the first periodic Bernoulli polynomial as a finite combination of cotangent values, as a relation between two bases of the vector space of periodic Dirichlet series. We shall also determine the limiting behavior of them, giving rise to Gauss’ famous closed formula for the values of the digamma function at rational points on the one hand and elucidation of Eisenstein-Wang’s formulas in the context of Kubert functions on the other.

W shall reveal that most of the relevant previous results are the combinations of the generalized Eisenstein formula and the functional equation.

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Authors and Affiliations

  1. Department of Mathematics, Weinan Teachers University, Weinan, 714000, China

    HaiLong Li

  2. Department of Mathematics, Wakayama College of Technology, Noda-machi Gobo-shi, 649-15, Japan

    Masahiro Hashimoto

  3. Graduate School of Advanced Technology, Kinki University Iizuka, Fukuoka, 820-8555, Japan

    Shigeru Kanemitsu

Authors
  1. HaiLong Li
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  2. Masahiro Hashimoto
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  3. Shigeru Kanemitsu
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Correspondence to HaiLong Li.

Additional information

Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

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Li, H., Hashimoto, M. & Kanemitsu, S. The structural elucidation of Eisenstein’s formula. Sci. China Math. 53, 2341–2350 (2010). https://doi.org/10.1007/s11425-010-3086-8

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