Abstract
B.C. Berndt (J. Reine Angew. Math. 272:182–193, 1975; 304:332–365, 1978) has derived a number of new transformation formulas, in particular, the transformation formulae of the logarithms of the classical theta functions, by using a transformation formula for a more general class of Eisenstein series. In this paper, we continue his study. By using a transformation formula for a class of twisted generalized Eisenstein series, we generalize a transformation formula given by J. Lehner (Duke Math. J. 8:631–655, 1941) and give a new proof for transformation formulas proved by Y. Yang (Bull. Lond. Math. Soc. 36:671–682, 2004).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1965)
Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partitions. Bull. Am. Math. Soc. (N.S.) 18, 167–171 (1988)
Berndt, B.C.: Two new proofs of Lerch’s functional equation. Proc. Am. Math. Soc. 32, 403–408 (1972)
Berndt, B.C.: Generalized Dedekind eta-functions and generalized Dedekind sums. Trans. Am. Math. Soc. 178, 495–508 (1973)
Berndt, B.C.: Generalized Eisenstein series and modified Dedekind sums. J. Reine Angew. Math. 272, 182–193 (1975)
Berndt, B.C.: Modular transformations and generalizations of several formulae of Ramanujan. Rocky Mt. J. Math. 7(1), 147–189 (1977)
Berndt, B.C.: Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math. 304, 332–365 (1978)
Kubert, D., Lang, S.: Modular Units. Grundlehren der mathematischen. Wissenschaften, vol. 244, Springer, New York (1981)
Lehner, J.: A partition function connected with the modulus five. Duke Math J. 8, 631–655 (1941)
Mahlburg, K.: Partition congruences and the Andrews-Garvan-Dyson crank. Proc. Natl. Acad. Sci. USA 102, 15373–15376 (2005)
Newman, M.: Construction and application of a class of modular functions. II. Proc. Lond. Math. Soc. 9(3), 373–387 (1959)
Rademacher, H.: Some remarks on certain generalized Dedekind sums. Acta Arith. 9, 97–105 (1964)
Ramanujan, S.: Proof of certain identities in combinatory analysis. Proc. Camb. Philos. Soc. 19, 214–216 (1919)
Ramanujan, S.: Collected Papers. Cambridge University Press, Cambridge (1927). Reprinted by Chelsey, New York (1962). Reprinted by the Am. Math. Soc., Providence (2000)
Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894)
Schur, I.: Zur Additiven Zahlentheorie. Sitzungsberichte der Berliner Akademie (1926), pp. 488–495.
Yang, Y.: Transformation formulas for generalized Dedekind eta functions. Bull. Lond. Math. Soc. 36, 671–682 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-214-C00003). This work also partially supported by BK21-Postech CoDiMaRo.
Rights and permissions
About this article
Cite this article
Lim, SG. Generalized Eisenstein series and several modular transformation formulae. Ramanujan J 19, 121–136 (2009). https://doi.org/10.1007/s11139-008-9146-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-008-9146-9