Jacobi triple product

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In mathematics, the Jacobi triple product is a relation that re-expresses the Jacobi theta function, normally written as a series, as a product. This relationship generalizes other results, such as the pentagonal number theorem.

Let x and y be complex numbers, with |x| < 1 and y not zero. Then

$\prod_{m=1}^\infty \left( 1 - x^{2m}\right) \left( 1 + x^{2m-1} y^2\right) \left( 1 + x^{2m-1} y^{-2}\right) = \sum_{n=-\infty}^\infty x^{n^2} y^{2n}.$

This can easily be seen to be a relation on the Jacobi theta function; taking $x = exp(i πτ)$ and $y = exp(i π z)$ one sees that the right hand side is

$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (i\pi n^2 \tau + 2i \pi n z)$ .

Euler's pentagonal number theorem follows by taking $x = q 3 / 2$ and $y^2=-\sqrt{q}$ . One then gets

$\phi(q) = \prod_{m=1}^\infty \left(1-q^m \right) = \sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.\,$

The Jacobi triple product enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function, which see. It also takes on a concise form when expressed in terms of q-series:

$\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n = (q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.$