On a class of elliptic functions associated with imaginary quadratic fields
HTML articles powered by AMS MathViewer
- by Li-Chien Shen PDF
- Proc. Amer. Math. Soc. 132 (2004), 463-471 Request permission
Abstract:
Let $-D$ be the field discriminant of an imaginary quadratic field. We construct a class of elliptic functions associated naturally with the quadratic field $Q(\sqrt {-D})$ which, combined with the general theory of elliptic functions, allows us to provide a unified theory for two fundamental results (one classical and one due to Ramanujan) about the elliptic functions.References
- A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
- Harvey Cohn, Advanced number theory, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1980. Reprint of A second course in number theory, 1962. MR 594936
- Bruce C. Berndt, S. Bhargava, and Frank G. Garvan, Ramanujan’s theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4163–4244. MR 1311903, DOI 10.1090/S0002-9947-1995-1311903-0
- Li-Chien Shen, On an identity of Ramanujan based on the hypergeometric series $_2F_1(\frac 13,\frac 23;\frac 12; x)$, J. Number Theory 69 (1998), no. 2, 125–134. MR 1617305, DOI 10.1006/jnth.1997.2212
- Li-Chien Shen, On a class of $q$-series related to quadratic forms, Bull. Inst. Math. Acad. Sinica 26 (1998), no. 2, 111–126. MR 1633743
- E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Cambridge University Press, New York, 1962. Fourth edition. Reprinted. MR 0178117
Additional Information
- Li-Chien Shen
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-2082
- Email: shen@math.ufl.edu
- Received by editor(s): October 3, 2002
- Published electronically: August 28, 2003
- Communicated by: David E. Rohrlich
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 463-471
- MSC (2000): Primary 33E05
- DOI: https://doi.org/10.1090/S0002-9939-03-07259-9
- MathSciNet review: 2022370