A theta function identity and its implications

Liu, Zhi-Guo

doi:10.1090/S0002-9947-04-03572-X

A theta function identity and its implications

Author: Zhi-Guo Liu
Journal: Trans. Amer. Math. Soc. 357 (2005), 825-835
MSC (2000): Primary 11F11, 11F12, 11F27, 33E05
DOI: https://doi.org/10.1090/S0002-9947-04-03572-X
Published electronically: September 2, 2004
MathSciNet review: 2095632
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for $\prod_{n=1}^\infty(1-q^n)^{10}$ is given. The proofs are self-contained and elementary.

1. G. E. Andrews, ``The Theory of Partitions,'' Addison-Wesly, Reading, MA, 1976. MR 58:27738
2. B. C. Berndt, ``Ramanujan's Notebooks, Part III,'' Springer-Verlag, New York, 1991. MR 92j:01069
3. B. C. Berndt, ``Ramanujan's notebooks. Part IV,'' Springer-Verlag, New York, 1994. MR 95e:11028
4. B. C. Berndt, On a certain theta-function in a letter of Ramanujan from Fitzroy House, Ganita 43 (1992), 33-43. MR 94b:11105
5. Bruce C. Berndt, Song Heng Chan, Zhi-Guo Liu, and Hamza Yesilyurt, A new identity for (𝑞;𝑞)¹⁰_{∞} with an application to Ramanujan’s partition congruence modulo 11, Q. J. Math. 55 (2004), no. 1, 13–30. MR 2043004, https://doi.org/10.1093/qjmath/55.1.13
6. K. Chandrasekharan, ``Elliptic Functions,'' Springer-Verlag, Berlin Heideberg, 1985. MR 87e:11058
7. M. D. Hirschhorn, ``A simple proof of an identity of Ramanujan,'' J. Austral. Math. Soc. Ser. A, 34 (1983), 31-35. MR 84h:10067
8. M. Hirschhorn, F. Garvan, and J. Borwein, Cubic analogues of the Jacobian theta function $\theta(z,q)$ . Canad. J. Math. 45 (1993), 673-694. MR 94m:33011
9. Z.-G. Liu, The Borweins' cubic theta function identity and some cubic modular identities of Ramanujan, Ramanujan J. 4 (2000), 43-50. MR 2001f:33026
10. Sarachai Kongsiriwong and Zhi-Guo Liu, Uniform proofs of 𝑞-series-product identities, Results Math. 44 (2003), no. 3-4, 312–339. MR 2028683, https://doi.org/10.1007/BF03322989
11. Z.-G. Liu, On certain identities of Ramanujan, J. Number Theory 83 (2000), 59-75. MR 2001f:11066
12. Z.-G. Liu, Some Eisenstein series identities, J. Number Theory 85 (2000), 231-252. MR 2001k:11075
13. Z.-G. Liu, Some theta function identities associated with the modular equations of degree 5, Integers: Electronic Journal of Combinatorial Number theory 1 (2001), A#03, 14pp. MR 2002c:33017
14. Z.-G. Liu, Residue theorem and theta function identities, Ramanujan J. 5 (2001), 129-151.MR 2002g:11056
15. Z.-G. Liu, Some Eisenstein series identities related to modular equations of the seventh order. Pacific J. Math. 209 (2003), 103-130. MR 2004c:11052
16. E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed, Cambridge Univ. Press, Cambridge, 1966. MR 97k:01072

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11F11, 11F12, 11F27, 33E05

Retrieve articles in all journals with MSC (2000): 11F11, 11F12, 11F27, 33E05

Additional Information

Zhi-Guo Liu
Affiliation: Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
Email: zgliu@math.ecnu.edu.cn, liuzg18@hotmail.com

DOI: https://doi.org/10.1090/S0002-9947-04-03572-X
Keywords: Elliptic functions, theta function identities, infinite products
Received by editor(s): September 18, 2003
Received by editor(s) in revised form: October 24, 2003
Published electronically: September 2, 2004
Additional Notes: The author was supported in part by Shanghai Priority Academic Discipline and the National Science Foundation of China
Article copyright: © Copyright 2004 American Mathematical Society