Jacobi triple product

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 $x=\exp(i\pi \tau )$ and $y=\exp(i\pi z)$ one sees that the right hand side is

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

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:

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

Here, $(a;q)_{n}$ is the q-series.

References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York ISBN 0-387-90163-9 See chapter 14, theorem 14.6.