Jacobi triple product

In mathematics, the Jacobi triple product is the identity:

:<math>\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},

</math>

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 A<sub>1</sub>, 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 <math>x=q\sqrt q</math> and <math>y^2=-\sqrt{q}</math>. Then we have

:<math>\phi(q) = \prod_{m=1}^\infty \left(1-q^m \right) =

\sum_{n=-\infty}^\infty (-1)^n q^{\frac{3n^2-n}{2.</math>

The Rogers–Ramanujan identities follow with <math>x=q^2\sqrt q</math>, <math>y^2=-\sqrt{q}</math> and <math>x=q^2\sqrt q</math>, <math>y^2=-q\sqrt{q}</math>.

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

Let <math>x=e^{i\pi \tau}</math> and <math>y=e^{i\pi z}.</math>

Then the Jacobi theta function

:<math>

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

</math>

can be written in the form

:<math>\sum_{n=-\infty}^\infty y^{2n}x^{n^2}. </math>

Using the Jacobi triple product identity, the theta function can be written as the product

:<math>\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].

</math>

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:

:<math>\sum_{n=-\infty}^\infty q^{\frac{n(n+1)}{2z^n =

(q;q)_\infty \; \left(-\tfrac{1}{z};q\right)_\infty \; (-zq;q)_\infty,</math>

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

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

:<math>\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.</math>

Proof

Let <math>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)</math>

Substituting for and multiplying the new terms out gives

:<math>f_x(xy) = \frac{1+x^{-1}y^{-2{1+xy^2}f_x(y) = x^{-1}y^{-2}f_x(y)</math>

Since <math>f_x</math> is meromorphic for <math>|y| > 0</math>, it has a Laurent series

:<math>f_x(y)=\sum_{n=-\infty}^\infty c_n(x)y^{2n}</math>

which satisfies

:<math>\sum_{n=-\infty}^\infty c_n(x)x^{2n+1} y^{2n}=x f_x(x y)=y^{-2}f_x(y)=\sum_{n=-\infty}^\infty c_{n+1}(x)y^{2n}</math>

so that

:<math>c_{n+1}(x) = c_n(x)x^{2n+1} = \dots = c_0(x) x^{(n+1)^2}</math>

and hence

:<math>f_x(y)=c_0(x) \sum_{n=-\infty}^\infty x^{n^2} y^{2n}</math>

Evaluating

To show that <math>c_0(x) = 1</math>, use the fact that the infinite expansion

:<math>\prod_{m=1}^\infty \left(1 + x^{2m-1} y^2\right)\left(1 +x^{2m-1}y^{-2}\right)</math>

has the following infinite polynomial coefficient at <math>y^0</math>

:<math>=\sum_{m=0}^\infty \frac{x^{2m^2{(1-x^2)^2(1-x^4)^2\cdots(1-x^{2m})^2}</math>

which is the Durfee square generating function with <math>x^2</math> instead of <math>x</math>.

:<math>=\prod_{m=1}^\infty \left(1 - x^{2m}\right)^{-1}</math>

Therefore at <math>y^0</math>we have <math>f_x(y)=1</math>, and so <math>c_0(x)=1</math>.

Other proofs

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

For the analytic case, see Apostol.

Jacobi triple product

Properties

Proof

Evaluating

Other proofs

References

Further reading