Jacobi triple product explained

In mathematics, the Jacobi triple product is the identity:

\prod ^{inf ty \left(} _{m =1} 1 - x ^{2m} \right) \left( 1 + x ^{2m-1} y ^{2\right) \left(} 1+ (x ^{2m-1})/(y ^{2}) \right) = \sum ^{inf ty} _{n =-inf ty} x ^{n ^{2}} y ^{2n} , 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 A₁, 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

x = q\sqrt q and y ^{2=-\sqrt{q}} . Then we have\phi(q) = \prod ^{inf ty} _{m =1} \left( 1- q ^{m} \right) = \sum ^{inf ty} _{n =-inf ty} ( -1 ) ^{n} q ^{(3 n ^{2-n})/(2)} .

The Rogers–Ramanujan identities follow with

x = q ^{2\sqrt} q , y ^{2=-\sqrt{q}} and x = q ^{2\sqrt} q , y ^{2=-q\sqrt{q}} .

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

Let

x = e ^{i\pi} and y = e ^{i\pi} .

Then the Jacobi theta function

\vartheta(z; \tau) = \sum ^{inf ty} _{n =-inf ty} e ^{\pi} } n^2 \tau + 2 \pi n z}

can be written in the form

\sum ^{inf ty} _{n =-inf ty} y ^{2n} x ^{n ^{2}} .

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

\vartheta(z; \tau) = \prod ^{inf ty \left(} _{m =1} 1 - e ^{2m} } \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 ^{inf ty} _{n =-inf ty} q ^{(n(n +1 ))/(2)} z ^{n} = (q;q) _{infty} \left( - \tfrac{ 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 ^{inf ty} _{n =-inf ty} a ^{(n(n +1 ))/(2)} b ^{(n(n -1 ))/(2)} = ( - a; ab) _{infty} ( - b; ab) _{infty} (ab;ab) _{infty.}

Proof

Let

f _{x(y)} = \prod ^{inf ty} _{m =1} \left( 1 - x ^{2m} \right)\left( 1 + x ^{2m-1} y ^{2\right)\left(} 1 + x ^{2m-1} y ^{-2} \right)

Substituting for and multiplying the new terms out gives

f _{x(xy)} = (1+ x ^{-1} y ^{-2})/(1+ xy ^{2}) f _{x(y)} = x ^{-1} y ^{-2} f _{x(y)}

Since

f _{x} is meromorphic for |y| > 0 , it has a Laurent seriesf _{x(y)=\sum} ^{inf ty} ^{n =-inf ty} c ^{2 n} _{n(x)y}

which satisfies

\sum ^{inf ty} _{n =-inf ty} c ^{2 n +1} _{n(x)x} y ^{2n} = x f _{x(x} y) = y ^{-2} f _{x(y)=\sum} ^{inf ty} ^{n =-inf ty} c _{n+1} (x)y ^{2n}

so that

c _{n+1} (x) = c ^{2 n +1} _{n(x)x} = ... = c _{0(x)} x ^{(n +1 ) ^{2}}

and hence

f _{x(y)=c} _{0(x)} \sum ^{inf ty} _{n =-inf ty} x ^{n ^{2}} y ^{2n}

Evaluating

Showing that

c _{0(x)} = 1 (the polynomial of x of y ^{0} is 1) is technical. One way is to set y = 1 and show both the numerator and the denominator of(1 {c)/(_{0(e}) ^{2i\pi} )} = (\sum\limits ^{inf ty} _{n =-inf ty} e ^{2 i\pi n ^{2} z})/(\prod\limits ^{inf ty} _{m =1} ( 1- e ^{2i\pi} )( 1+ e ^{2i\pi(2m-1)z} ) ^{2})

are weight 1/2 modular under

z\mapsto - (1)/(4 z) , since they are also 1-periodic and bounded on the upper half plane the quotient has to be constant so that c _{0(x)=c} _{0(0)=1} .

Other proofs

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

For the analytic case, see Apostol.^[2]

References

• Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, (1994) Cambridge University Press,

Notes and References

1. Andrews. George E.. 1965-02-01. A simple proof of Jacobi's triple product identity. Proceedings of the American Mathematical Society. en-US. 16. 2. 333. 10.1090/S0002-9939-1965-0171725-X. 0002-9939. free.
2. Chapter 14, theorem 14.6 of