Quintuple Product Identity

A.k.a. the Watson Quintuple Product Identity.

$\prod_{n=1}^\infty (1-q^n)(1-zq^n)(1-z^{-1}q^{n-1})(1-z^2q^{2n-1})$

$\times(1-z^{-2}q^{2n-1}) = \sum_{m=-\infty}^\infty (z^{3m}-z^{-3m-1})q^{m(2m+1)/2}.$

(1)

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)

or

$\sum_{k=-\infty}^\infty (-1)^kq^{(3k^2-k)/2}x^{3k}(1+zq^k)$

$=\prod_{j=1}^\infty (1-q^j)(1+z^{-1}q^j)(1+zq^{j-1})(1+z^{-2}q^{2j-1})(1+z^2q^{2j-1}).\quad$ (3)

Using the Notation of the Ramanujan Theta Function (Berndt, p. 83),

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

(4)

References

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.

$\prod_{n=1}^\infty (1-q^n)(1-zq^n)(1-z^{-1}q^{n-1})(1-z^2q^{2n-1})$
$\times(1-z^{-2}q^{2n-1}) = \sum_{m=-\infty}^\infty (z^{3m}-z^{-3m-1})q^{m(2m+1)/2}.$
	(1)

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