On the modular equations of degree

Author:
Li-Chien Shen

Journal:
Proc. Amer. Math. Soc. **122** (1994), 1101-1114

MSC:
Primary 11F27; Secondary 33E05

MathSciNet review:
1212287

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we use the twelve Jacobian elliptic functions to derive a collection of 24 identities which are essential in the study of the modular equations of degree 3.

**[1]**Bruce C. Berndt,*Ramanujan’s notebooks. Part III*, Springer-Verlag, New York, 1991. MR**1117903****[2]**-,*Ramanujan's theory of theta-functions*, preprint.**[3]**J. M. Borwein and P. B. Borwein,*Pi and the AGM--A study in analytic number theory and computational complexity*, Wiley, New York, 1990.**[4]**J. M. Borwein and P. B. Borwein,*A cubic counterpart of Jacobi’s identity and the AGM*, Trans. Amer. Math. Soc.**323**(1991), no. 2, 691–701. MR**1010408**, 10.1090/S0002-9947-1991-1010408-0**[5]**J. M. Borwein, P. B. Borwein, and F. G. Garvan,*Some cubic modular identities of Ramanujan*, Trans. Amer. Math. Soc.**343**(1994), no. 1, 35–47. MR**1243610**, 10.1090/S0002-9947-1994-1243610-6**[6]**G. H. Hardy,*Ramanujan*, 3rd ed., Chelsea, New York, 1978.**[7]**E. T. Whittaker and G. N. Watson,*A course of modern analysis*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR**1424469**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1212287-3

Keywords:
Theta functions,
Jacobian elliptic functions,
Lambert series,
modular identity

Article copyright:
© Copyright 1994
American Mathematical Society