Jacobi triple product

Short description: Mathematical identity found by Jacobi in 1829

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

[math]\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}, }[/math]

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 [math]\displaystyle{ x=q\sqrt q }[/math] and [math]\displaystyle{ y^2=-\sqrt{q} }[/math]. Then we have

[math]\displaystyle{ \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 Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows:

Let [math]\displaystyle{ x=e^{i\pi \tau} }[/math] and [math]\displaystyle{ y=e^{i\pi z}. }[/math]

Then the Jacobi theta function

[math]\displaystyle{ \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]\displaystyle{ \sum_{n=-\infty}^\infty y^{2n}x^{n^2}. }[/math]

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

[math]\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]. }[/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]\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, }[/math]

where [math]\displaystyle{ (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]\displaystyle{ |ab|\lt 1 }[/math] it can be written as

[math]\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. }[/math]

Proof

Let [math]\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) }[/math]

Substituting xy for y and multiplying the new terms out gives

[math]\displaystyle{ 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]\displaystyle{ f_x }[/math] is meromorphic for [math]\displaystyle{ |y| \gt 0 }[/math], it has a Laurent series

[math]\displaystyle{ f_x(y)=\sum_{n=-\infty}^\infty c_n(x)y^{2n} }[/math]

which satisfies

[math]\displaystyle{ \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]\displaystyle{ c_{n+1}(x) = c_n(x)x^{2n+1} = \dots = c_0(x) x^{(n+1)^2} }[/math]

and hence

[math]\displaystyle{ f_x(y)=c_0(x) \sum_{n=-\infty}^\infty x^{n^2} y^{2n} }[/math]

Evaluating c₀(x)

Showing that [math]\displaystyle{ c_0(x) =1 }[/math] is technical. One way is to set [math]\displaystyle{ y= 1 }[/math] and show both the numerator and the denominator of

[math]\displaystyle{ \frac1{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} }[/math]

are weight 1/2 modular under [math]\displaystyle{ z\mapsto -\frac{1}{4z} }[/math], since they are also 1-periodic and bounded on the upper half plane the quotient has to be constant so that [math]\displaystyle{ c_0(x)=c_0(0)=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.

References

• Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, (1994) Cambridge University Press , ISBN:0-521-45761-0
• Jacobi, C. G. J. (1829) (in Latin), Fundamenta nova theoriae functionum ellipticarum, Königsberg: Borntraeger, Reprinted by Cambridge University Press 2012, ISBN 978-1-108-05200-9, https://archive.org/details/fundamentanovat00jacogoog 
• Carlitz, L (1962), A note on the Jacobi theta formula, American Mathematical Society, https://projecteuclid.org/euclid.bams/1183524930 
• Wright, E. M. (1965), "An Enumerative Proof of An Identity of Jacobi", Journal of the London Mathematical Society (London Mathematical Society): 55–57, doi:10.1112/jlms/s1-40.1.55