In mathematics, the Jacobi triple product is the identity:

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

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Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi triple product identity.

Let   and  . Then we have

 

The Rogers–Ramanujan identities follow with  ,   and  ,  .

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

Let   and  

Then the Jacobi theta function

 

can be written in the form

 

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

 

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:

 

where   is the infinite q-Pochhammer symbol.

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

 

Proof

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Let  

Substituting xy for y and multiplying the new terms out gives

 

Since   is meromorphic for  , it has a Laurent series

 

which satisfies

 

so that

 

and hence

 

Evaluating c0(x)

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Showing that   (the polynomial of x of   is 1) is technical. One way is to set   and show both the numerator and the denominator of

 

are weight 1/2 modular under  , since they are also 1-periodic and bounded on the upper half plane the quotient has to be constant so that  .

Other proofs

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

For the analytic case, see Apostol.[2]

References

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  1. ^ Andrews, George E. (1965-02-01). "A simple proof of Jacobi's triple product identity". Proceedings of the American Mathematical Society. 16 (2): 333. doi:10.1090/S0002-9939-1965-0171725-X. ISSN 0002-9939.
  2. ^ 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