Jacobi triple product

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

${\displaystyle \prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\right)\left(1+{\frac {x^{2m-1}}{y^{2}}}\right)=\sum _{n=-\infty }^{\infty }x^{n^{2}}y^{2n},}$

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.

Properties

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

${\displaystyle \phi (q)=\prod _{m=1}^{\infty }\left(1-q^{m}\right)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{\frac {3n^{2}-n}{2}}.}$

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

Let ${\displaystyle x=e^{i\pi \tau }}$  and ${\displaystyle y=e^{i\pi z}.}$

Then the Jacobi theta function

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

can be written in the form

${\displaystyle \sum _{n=-\infty }^{\infty }y^{2n}x^{n^{2}}.}$

Using the Jacobi Triple Product Identity we can then write the theta function as the product

${\displaystyle \vartheta (z;\tau )=\prod _{m=1}^{\infty }\left(1-e^{2m\pi {\rm {i}}\tau }\right)\left[1+e^{(2m-1)\pi {\rm {i}}\tau +2\pi {\rm {i}}z}\right]\left[1+e^{(2m-1)\pi {\rm {i}}\tau -2\pi {\rm {i}}z}\right].}$

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:

${\displaystyle \sum _{n=-\infty }^{\infty }q^{\frac {n(n+1)}{2}}z^{n}=(q;q)_{\infty }\;\left(-{\tfrac {1}{z}};q\right)_{\infty }\;(-zq;q)_{\infty },}$

where ${\displaystyle (a;q)_{\infty }}$  is the infinite q-Pochhammer symbol.

It enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For ${\displaystyle |ab|<1}$  it can be written as

${\displaystyle \sum _{n=-\infty }^{\infty }a^{\frac {n(n+1)}{2}}\;b^{\frac {n(n-1)}{2}}=(-a;ab)_{\infty }\;(-b;ab)_{\infty }\;(ab;ab)_{\infty }.}$

Proof

Let ${\displaystyle f_{x}(y)=\prod _{m=1}^{\infty }\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\right)\left(1+x^{2m-1}y^{-2}\right)}$

Substituting xy for y and multiplying the new terms out gives

${\displaystyle f_{x}(xy)={\frac {1+x^{-1}y^{-2}}{1+xy^{2}}}f_{x}(y)}$

and, because ${\displaystyle \textstyle 1+{\frac {1}{z}}={\frac {1}{z}}(1+z)}$ ,

${\displaystyle f_{x}(xy)=x^{-1}y^{-2}f_{x}(y)}$

Since fx is meromorphic for ${\displaystyle |y|>0}$ , it has a Laurent series

${\displaystyle f_{x}(y)=\sum _{n=-\infty }^{\infty }c_{n}(x)y^{2n}}$

which satisfies

${\displaystyle \sum _{n=-\infty }^{\infty }c_{n}(x)x^{2n+1}y^{2n}=xf_{x}(xy)=y^{-2}f_{x}(y)=\sum _{n=-\infty }^{\infty }c_{n+1}(x)y^{2n}}$

so that

${\displaystyle c_{n+1}(x)=c_{n}(x)x^{2n+1}=\dots =c_{0}(x)x^{(n+1)^{2}}}$

and hence

${\displaystyle f_{x}(y)=c_{0}(x)\sum _{n=-\infty }^{\infty }x^{n^{2}}y^{2n}}$

A different proof is given by G. E. Andrews based on two identities of Euler.[1]

For the analytic case, see Apostol.[2]

Evaluating ${\displaystyle c_{0}(x)}$

Evaluating ${\displaystyle c_{0}(x)}$  is more technical. One method is to set y=1 and show both the numerator and the denominator of

${\displaystyle {\frac {1}{c_{0}(e^{2i\pi z})}}={\frac {\sum \limits _{n=-\infty }^{\infty }e^{2i\pi n^{2}z}}{\prod \limits _{m=1}^{\infty }(1-e^{2i\pi mz})(1+e^{2i\pi (2m-1)z})^{2}}}}$

are weight 1/2 modular under ${\displaystyle z\mapsto -{\frac {1}{4z}}}$ , since they are also 1-periodic and bounded on the upper half plane the quotient has to be constant so that ${\displaystyle c_{0}(x)=c_{0}(0)=1}$ .

References

1. ^ Andrews, George E. (1965-02-01). "A simple proof of Jacobi's triple product identity". Proceedings of the American Mathematical Society (in American English). 16 (2): 333. doi:10.1090/S0002-9939-1965-0171725-X. ISSN 0002-9939.
2. ^ Chapter 14, theorem 14.6 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001