Analogues of the Hurwitz Formulas for Level 2 Eisenstein Series

被引：0
作者：
Hirofumi Tsumura
机构：
[1] Tokyo Metropolitan University,Department of Mathematics and Information Sciences
来源：
Results in Mathematics | 2010年 / 58卷
关键词：
Primary 11M41; Secondary 11M99; Eisenstein series; Hurwitz numbers; hyperbolic functions; Lemniscate constant; Riemann zeta-function;
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摘要：
In this paper, we consider certain double series of Eisenstein type involving hyperbolic functions, which can be regarded as analogues of the level 2 Eisenstein series. We prove some evaluation formulas for these series at positive integers which are analogues of both the Hurwitz formulas for the level 2 Eisenstein series and the classical results given by Cauchy, Lerch, Mellin and Ramanujan.
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页码：365 / 378
页数：13
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共 16 条
  • [1] Ayoub R.(1974)Euler and the zeta function Am. Math. Mon. 81 1067-1086
  • [2] Berndt B.C.(1977)Modular transformations and generalizations of several formulae of Ramanujan Rocky Mt. J. Math. 7 147-189
  • [3] Berndt B.C.(1978)Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan J. Reine Angew Math. 303/304 332-365
  • [4] Katayama K.(1978)On the values of Eisenstein series Tokyo J. Math. 1 157-188
  • [5] Lerch M.(1901)Sur la fonction ζ ( J. Sci. Math. Astron. pub. pelo Dr. F. Gomes Teixeira, Coimbra 14 65-69
  • [6] Mellin H.J.(1902)) pour valeurs impaires de l’argument Acta Soc. Sci. Fennicae 29 49-183
  • [7] Mellin H.J.(1902)Eine Formel für den Logarithmus transcendenter Funktionen von endlichem Geschlecht Acta. Math. 25 165-138
  • [8] Malurkar S.L.(1925/26)Eine Formel für den Logarithmus transcendenter Funktionen von endlichem Geschlecht J. Indian Math. Soc. 16 130-228
  • [9] Nanjundiah T.S.(1951)On the application of Herr Mellin’s integrals to some series Proc. Indian Acad. Sci. Sec. A 34 215-116
  • [10] Phillips E.G.(1929)Certain summations due to Ramanujan, and their generalizations J. Lond. Math. Soc. 4 114-216
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