Quintuple Product Identity
The quintuple product identity, also called the Watson quintuple product identity, states
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(1)
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It can also be written
![product_(n=1)^infty(1-q^(2n))(1-q^(2n-1)z)(1-q^(2n-1)z^(-1))(1-q^(4n-4)z^2)(1-q^(4n-4)z^(-2))
=sum_(n=-infty)^inftyq^(3n^2-2n)[(z^(3n)+z^(-3n))-(z^(3n-2)+z^(-(3n-2)))]](http://mathworld.wolfram.com/images/equations/QuintupleProductIdentity/NumberedEquation2.gif) |
(2)
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or
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(3)
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The quintuple product identity can be written in q-series notation as
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(4)
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where 
and 
(Gasper and Rahman 1990, p. 134; Leininger and Milne 1999).
Using the notation of the Ramanujan theta function (Berndt 1985, p. 83),
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(5)
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REFERENCES:
Berndt, B. C. Ramanujan's Notebooks, Part III. New York:Springer-Verlag, 1985.
Bhargava, S. "A Simple Proof of the Quintuple Product Identity." J. Indian Math. Soc. 61, 226-228, 1995.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 306-309, 1987.
Carlitz, L. and Subbarao, M. V. "A Simple Proof of the Quintuple Product Identity." Proc. Amer. Math. Soc. 32, 42-44, 1972.
Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.
Leininger, V. E. and Milne, S. C. "Some New Infinite Families of 
-Function Identities." Methods Appl. Anal. 6, 225-248, 1999b.
