Jacobi triple product

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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.

Properties[edit]

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

Proof[edit]

Let then . Since fx is meromorphic for |y| > 0 it has a Laurent series which satisfies so that and hence

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 .

A simple proof is given by G. E. Andrews based on two identities of Euler.[1] For the analytic case, see Apostol, the first edition of which was published in 1976. Also see links below for a proof motivated with physics due to Borcherds[citation needed].

References[edit]

  • See 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
  • Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, (1994) Cambridge University Press, ISBN 0-521-45761-0
  • Jacobi, C. G. J. (1829), Fundamenta nova theoriae functionum ellipticarum (in Latin), Königsberg: Borntraeger, ISBN 978-1-108-05200-9, Reprinted by Cambridge University Press 2012
  • Carlitz, L (1962), A note on the Jacobi theta formula, American Mathematical Society
  • 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
  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.

External links[edit]