Quintuple Product Identity

The quintuple product identity, also called the Watson quintuple product identity, states

(1)

It can also be written

(2)

or

(3)

The quintuple product identity can be written in q-series notation as

(4)

where  and  (Gasper and Rahman 1990, p. 134; Leininger and Milne 1999).

Using the notation of the Ramanujan theta function (Berndt 1985, p. 83),

(5)

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.