Jacobi triple product

In mathematics, the Jacobi triple product is the identity:
for complex numbers x and y, with |x| < 1 and y ≠ 0. It was introduced by 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.

Properties

Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi triple product identity.
Let and. Then we have
The Rogers–Ramanujan identities follow with, and,.
The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows:
Let and
Then the Jacobi theta function
can be written in the form
Using the Jacobi triple product identity, the theta function can be written as the product
There are many different notations used to express the Jacobi triple product. It takes on a concise form when expressed in terms of q-Pochhammer symbols:
where is the infinite q-Pochhammer symbol.
It enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For it can be written as

Let
Substituting for and multiplying the new terms out gives
Since is meromorphic for, it has a Laurent series
which satisfies
so that
and hence

To show that, use the fact that the infinite expansion
has the following infinite polynomial coefficient at
which is the Durfee square generating function with instead of.
Therefore at we have, and so.

A different proof is given by G. E. Andrews based on two identities of Euler.
For the analytic case, see Apostol.