Jacobi triple product

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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 $x = exp(i πτ)$ and $y = exp(i π z)$ one sees that the right hand side is

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

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-Pochhammer symbols:

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

Here, $(a;q)_\infty$ is the infinite q-Pochhammer symbol.

[edit] 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 $\sum_{m,l} ?\,q^mz^l$ for the number 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, ${v : v < l}$ , 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 $λ 1$ levels, the next highest particle up by $λ 2$ 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.

[edit] 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.
Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, (1994) Cambridge University Press, ISBN 0-521-45761-0