The Ramanujan Journal

, Volume 42, Issue 2, pp 479–489 | Cite as

On multiple series of Eisenstein type



The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As part of a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as polynomials in terms of ordinary Eisenstein series. This fact was derived from the analytic observation of infinite series involving hyperbolic functions which were based on the study of Cauchy, and also Ramanujan. In this paper, we prove an explicit relation formula among these series. This gives an alternative proof of this fact by using the technique of partial fraction decompositions of multiple series which was introduced by Gangl, Kaneko and Zagier. By the same method, we further show a certain multiple analogue of this fact and give some examples of explicit formulas. Finally we give several remarks about the relation between the results of the present and the previous works for infinite series involving hyperbolic functions.


Multiple Eisenstein series Riemann zeta function   Hyperbolic functions Lemniscate constant 

Mathematics Subject Classification

11M41 11M99 


  1. 1.
    Berndt, B.C.: Ramanujan’s Notebooks. Part II. Springer-Verlag, New York (1989)MATHGoogle Scholar
  2. 2.
    Cauchy, A.L.: Exercices de Mathématiques, Paris, 1827; Oeuvres Completes D’Augustin Cauchy, Série II, t. VII. Gauthier-Villars, Paris (1889)Google Scholar
  3. 3.
    Gangl, H., Kaneko, M., Zagier, D.: Double zeta values and modular forms. In: Automorphic Forms and Zeta Functions, pp. 71–106. World Scientific Publication, Hackensack (2006)Google Scholar
  4. 4.
    Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97, 2nd edn. Springer-Verlag, New York (1993)CrossRefMATHGoogle Scholar
  5. 5.
    Lemmermeyer, F.: Reciprocity Laws: From Euler to Eisenstein. Springer-Verlag, New York (2000)CrossRefMATHGoogle Scholar
  6. 6.
    Pasles, P.C., Pribitkin, W.A.: A generalization of the Lipschitz summation formula and some applications. Proc. Am. Math. Soc. 129, 3177–3184 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Serre, J.-P.: A Course in Arithmetic. Graduate Texts in Mathematics, vol. 7. Springer-Verlag, New York (1973)MATHGoogle Scholar
  8. 8.
    Tsumura, H.: On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type. Bull. Lond. Math. Soc. 40, 85–93 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Tsumura, H.: Evaluation of certain classes of Eisenstein’s type series. Bull. Aust. Math. Soc. 79, 239–247 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Tsumura, H.: Analogues of level-\(N\) Eisenstein series. Pac. J. Math. 255, 489–510 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversität HamburgHamburgGermany
  2. 2.Department of Mathematics and Information SciencesTokyo Metropolitan UniversityHachiojiJapan

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