Korean J. Math. Vol. 21 No. 3 (2013) pp.285-292
DOI: https://doi.org/10.11568/kjm.2013.21.3.285

Analytic continuation of generalized non-holomorphic Eisenstein series

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Published: Sep 23, 2013
Mathematics Subject Classification:
Primary 34A25, Secondry 11M36

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Sung-Geun Lim
Mokwon University

Abstract

B. C. Berndt computed the Fourier series of a class of generalized Eisenstein series, which gives an analytic continuation to the generalized Eisenstein series.

In this paper, continuing his work, we consider generalized non-holomorphic Eisenstein series and give an analytic continuation to the $s$-plane.



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References

[1] bibitem{B0} B. C. Berndt, {it Two new proofs of Lerch's functional equation}, Google Scholar

[2] Proc. Amer. Math. Soc. {bf 32} (1972), 403--408. Google Scholar

[3] bibitem{B2} B. C. Berndt, {it Generalized Eisenstein series and modified Dedekind sums}, Google Scholar

[4] J. Reine. Angew. Math. {bf 272} (1975), 182--193. Google Scholar

[5] bibitem{B4} B. C. Berndt, {it Modular transformations and generalizations of several formulae of Ramanujan}, Google Scholar

[6] The Rocky mountain J. Math. {bf 7}, no 1 (1977), 147--189. Google Scholar

[7] bibitem{B3} B. C. Berndt, {it Analytic Eisenstein series, theta-functions, and series relations Google Scholar

[8] in the spirit of Ramanujan}, J. Reine. Angew. Math. {bf 304} (1978), 332--365. Google Scholar

[9] bibitem{KN} M. Katsurada and T. Noda, {it Differential actions on the asymptotic expansions Google Scholar

[10] of non-holomorphic Eienstein series}, to appear. Google Scholar

[11] bibitem{Leb} N. N. Lebedev, {it Special functions and their applications}, Dover Publications, Inc., New York, 1972. Google Scholar

[12] bibitem{Li1} S. Lim, {it Generalized Eisenstein series and several modular transformation formulae}, Ramanujan J. {bf 19} (2009), 121--136. Google Scholar

[13] bibitem{Li2} S. Lim, {it Modular transformation formulae for non-holomorphic Eisenstein series}, to appear. Google Scholar

[14] bibitem{PP} P. Pasles and W. Pribitkin, {it A generalization of the Lipschitz summation formula and applications}, Google Scholar

[15] Proc. Ams. Math. Soc. {bf 129}, no 11 (2001), 3177--3184. Google Scholar

[16] bibitem{Sl} L. J. Slater, {it Confluent hypergeometric functions}, Cambridge University Press, 1960. Google Scholar

[17] bibitem{Ti} E. C. Titchmarsh, {it Theory of functions}, Oxford University Press, 1952. Google Scholar