Jacobi triple product

In mathematics, the Jacobi triple product is the mathematical identity:

for complex numbers x and y, with |x| < 1 and y ≠ 0.

It was introduced by Jacobi (1829) in his work Fundamenta Nova Theoriae Functionum Ellipticarum.

The Jacobi triple product identity is the Macdonald identity for the affine root system of type A₁, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra.

The basis of Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product Identity.

Let ${\displaystyle x=q{\sqrt {q}}}$ and ${\displaystyle y^{2}=-{\sqrt {q}}}$. Then we have

The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows:

...
Wikipedia