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The Products of Three Theta Functions and the General Cubic Theta Functions

The Products of Three Theta Functions and the General Cubic Theta Functions
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摘要 In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities. In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.
作者 Xiao Mei YANG
机构地区 Department of Mathematics
出处 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2010年第6期1115-1124,共10页 数学学报（英文版）
基金 Supported by Innovation Program of Shanghai Municipal Education Commission and PCSIRT
关键词 Jacobi theta function quintuple product identity Winquist＇s identity Ramanujan identity Jacobi theta function, quintuple product identity, Winquist＇s identity, Ramanujan identity
分类号 O156.4 [理学—基础数学] TU247 [建筑科学—建筑设计及理论]
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  • 1Whittaker, E. T., Watson, G. N.: A Course of Mordern Analysis, 4th ed. (reprinted), Cambridge Univ. Press, Cambridge, 1962.
  • 2Hirschhorn, M. D., Garvan, F., Borwein, J.: Cubic analogues of the Jacobian theta function 0 (z, q). Canad. J. Math., 4,5, 473-694 (1993).
  • 3Borwein, J. M., Borwein, P. B., Garvan, F. G.: Some cubic modular identites of Ramanujan. Trans. Amer. Math. Soc., 343, 35-47 (1994).
  • 4Borwein, J. M., Borwein, P. B.: A cubic counterpart of Jacobi's identity and AGM. Trans. Amer. Math. Soc., 323, 691-701 (1991).
  • 5Ramanujian, S.: Collected Paper, Chelsea, New York, 1962.
  • 6Ramanujian, S.: Notebooks, Volume 2, TIFR, Bombay, 1957.
  • 7Berndt, B. c., Bhargava, S., Garvan, F. G.: Ramanujan's theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc., 347, 4163-4244 (1995).
  • 8Lewis, R., Liu, Z. -G.: The Borweins' cubic theta functions and q-elliptic functions, in: Symbolic Compu- tation, Number Theory, Special Functions, Physics and Combinatorics, F. G. Garvan and M. E. H. Ismail (Eds.), Kluwer Acad. Publ., Dordrecht, 2001, 133-145.
  • 9Bhargava, S.: Unification of the cubic analogues of the Jacobian theta-function. J. Math. Anal. Appl., 193, 543-558 (1995).
  • 10Chapman, It.: Cubic identities for theta series in three variables. Ramanujan J., 8, 459-465 (2005).
Acta Mathematica Sinica,English Series
2010年 第6期

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