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Quintuple Product Identity

A.k.a. the Watson Quintuple Product Identity.
$\prod_{n=1}^\infty (1-q^n)(1-zq^n)(1-z^{-1}q^{n-1})(1-z^2q^{2n-1})$
$ \times(1-z^{-2}q^{2n-1}) = \sum_{m=-\infty}^\infty (z^{3m}-z^{-3m-1})q^{m(2m+1)/2}.$

It can also be written

$\prod_{n=1}^\infty (1-q^{2n})(1-q^{2n-1}z)(1-q^{2n-1}z^{-1})(1-q^{4n-3}z^2)(1-q^{4n-4}z^{-2})$
$=\sum_{n=-\infty}^\infty q^{3n^2-2n}[(z^{3n}+z^{-3n})-(z^{3n-2}+z^{-(3n-2)})]\quad$ (2)

$\sum_{k=-\infty}^\infty (-1)^kq^{(3k^2-k)/2}x^{3k}(1+zq^k)$
$ =\prod_{j=1}^\infty (1-q^j)(1+z^{-1}q^j)(1+zq^{j-1})(1+z^{-2}q^{2j-1})(1+z^2q^{2j-1}).\quad$ (3)
Using the Notation of the Ramanujan Theta Function (Berndt, p. 83),

f(B^3/q,q^5/B^3) -B^2f(q/B^3,B^3q^5)=f(-q^2){f(-B^2,-q^2/B^2)\over f(Bq,q/B)}.
\end{displaymath} (4)

See also Jacobi Triple Product, Ramanujan Theta Functions


Berndt, B. C. Ramanujan's Notebooks, Part III. New York: Springer-Verlag, 1985.

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.

Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.

© 1996-9 Eric W. Weisstein