Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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IDENTITIES ABOUT LEVEL 2 EISENSTEIN SERIES

  • Xu, Ce (School of Mathematical Sciences Xiamen University)
  • Received : 2018.12.20
  • Accepted : 2019.05.16
  • Published : 2020.01.31
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Abstract

In this paper we consider certain classes of generalized level 2 Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formulas for some level 2 Eisenstein series. We can find that these level 2 Eisenstein series are reducible to infinite series involving hyperbolic functions. Moreover, some interesting new examples are given.

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References

  1. R. Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067-1086. https://doi.org/10.2307/2319041
  2. B. C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mountain J. Math. 7 (1977), no. 1, 147-189. https://doi.org/10.1216/RMJ-1977-7-1-147
  3. B. C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, J. Reine Angew. Math. 303/304 (1978), 332-365. https://doi.org/10.1515/crll.1978.303-304.332
  4. B. C. Berndt, Ramanujan's Notebooks. Part II, Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4612-4530-8
  5. B. C. Berndt, Ramanujan's Notebooks. Part III, Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0965-2
  6. G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, Collected papers of Srinivasa Ramanujan, Cambridge University Press, 1927,
  7. A. Hurwitz, Mathematische Werke. Bd. I, Herausgegeben von der Abteilung fur Mathematik und Physik der Eidgenossischen Technischen Hochschule in Zurich, Birkhauser Verlag, Basel, 1962.
  8. Y. Komori, K. Matsumoto, and H. Tsumura, Infinite series involving hyperbolic functions, Lith. Math. J. 55 (2015), no. 1, 102-118. https://doi.org/10.1007/s10986-015-9268-x
  9. C. B. Ling, On summation of series of hyperbolic functions, SIAM J. Math. Anal. 5 (1974), 551-562. https://doi.org/10.1137/0505055
  10. C. B. Ling, On summation of series of hyperbolic functions. II, SIAM J. Math. Anal. 6 (1975), 129-139. https://doi.org/10.1137/0506013
  11. H. Tsumura, On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type, Bull. Lond. Math. Soc. 40 (2008), no. 2, 289-297. https://doi.org/10.1112/blms/bdn014
  12. H. Tsumura, Evaluation of certain classes of Eisenstein-type series, Bull. Aust. Math. Soc. 79 (2009), no. 2, 239-247. https://doi.org/10.1017/S0004972708001159
  13. H. Tsumura, Analogues of the Hurwitz formulas for level 2 Eisenstein series, Results Math. 58 (2010), no. 3-4, 365-378. https://doi.org/10.1007/s00025-010-0058-9
  14. H. Tsumura, Analogues of level-N Eisenstein series, Pacific J. Math. 255 (2012), no. 2, 489-510. https://doi.org/10.2140/pjm.2012.255.489
  15. H. Tsumura, Double series identities arising from Jacobi's identity of the theta function, Results Math. 73 (2018), no. 1, Art. 10, 12 pp. https://doi.org/10.1007/s00025-018-0770-4
  16. C. Xu, Some evaluation of infinite series involving trigonometric and hyperbolic functions, Results Math. 73 (2018), no. 4, Art. 128, 18 pp. https://doi.org/10.1007/s00025-018-0891-9