Jacobi triple product

In mathematics, the Jacobi triple product is the mathematical 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.

PropertiesEdit

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   and  . Then we have

 

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 we can then write the theta function 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

 

ProofEdit

Let  

Substituting xy for y and multiplying the new terms out gives

 

and, because  ,

 

Since fx is meromorphic for  , it has a Laurent series

 

which satisfies

 

so that

 

and hence

 

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

For the analytic case, see Apostol.[2]

Evaluating Edit

Evaluating   is more technical. One method is to set y=1 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  .

ReferencesEdit

  1. ^ Andrews, George E. (1965-02-01). "A simple proof of Jacobi's triple product identity". Proceedings of the American Mathematical Society (in American English). 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