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http://dx.doi.org/10.4134/CKMS.c180528

IDENTITIES ABOUT LEVEL 2 EISENSTEIN SERIES  

Xu, Ce (School of Mathematical Sciences Xiamen University)
Publication Information
Communications of the Korean Mathematical Society / v.35, no.1, 2020 , pp. 63-81 More about this Journal
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.
Keywords
Eisenstein series; trigonometric function; hyperbolic function; Gamma function;
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1 R. Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067-1086. https://doi.org/10.2307/2319041   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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   DOI
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