Jacobi triple product

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

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

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 \(x=q\sqrt q \) and \(y^2=-\sqrt{q \)}. Then we have

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

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

Let \(x=e^{i\pi \tau} and y=e^{i\pi z}. \)

Then the Jacobi theta function

\( \vartheta(z; \tau) = \sum_{n=-\infty}^\infty e^{\pi {\rm{i}} n^2 \tau + 2 \pi {\rm{i}} n z} \)

can be written in the form

\( \sum_{n=-\infty}^\infty y^{2n}x^{n^2}. \)

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

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

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:

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

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

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

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

Proof

This proof uses a simplified model of the Dirac sea and follows the proof in Cameron (13.3) which is attributed to Richard Borcherds. It treats the case where the power series are formal. For the analytic case, see Apostol. The Jacobi triple product identity can be expressed as

\( \prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})=\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\prod_{n>0}(1-q^n)^{-1}\right). \)

A level is a half-integer. The vacuum state is the set of all negative levels. A state is a set of levels whose symmetric difference with the vacuum state is finite. The energy of the state S is

\( \sum\{v\colon v > 0,v\in S\} - \sum\{v\colon v < 0, v\not\in S\} \)

and the particle number of S is

\( |\{v\colon v>0,v\in S\}|-|\{v\colon v<0,v\not\in S\}|. \)

An unordered choice of the presence of finitely many positive levels and the absence of finitely many negative levels (relative to the vacuum) corresponds to a state, so the generating function \(\textstyle\sum_{m,l} s(m,l)q^mz^l \) for the number s(m,l) of states of energy m with l particles can be expressed as

\( \prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1}). \)

On the other hand, any state with l particles can be obtained from the lowest energy l-particle state, \(\{v\colon v<l\} \), by rearranging particles: take a partition \(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_j \) of m' and move the top particle up by \(\lambda_1 \) levels, the next highest particle up by \(\lambda_2 \) levels, etc.... The resulting state has energy \(m'+\frac{l^2}{2} \), so the generating function can also be written as

\(\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\sum_{n\geq0}p(n)q^n\right)=\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\prod_{n>0}(1-q^n)^{-1}\right) \)

where p(n) is the partition function. The uses of random partitions by Andrei Okounkov contains a picture of a partition exciting the vacuum.
Notes

References

See 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
Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, (1994) Cambridge University Press, ISBN 0-521-45761-0
Jacobi, C. G. J. (1829), Fundamenta nova theoriae functionum ellipticarum (in Latin), Königsberg: Borntraeger, ISBN 978-1-108-05200-9, Reprinted by Cambridge University Press 2012