“Jacobi triple product”的意思、由来-开放百科全书

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Jacobi triple product

释义

Properties
Proof
References
External links

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 {{harvs|txt|authorlink=Carl Gustav Jacob Jacobi|last=Jacobi|year=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 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

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|reason=The proof is basically the same as a well-known bijective proof of the identity which you can find in many textbooks. I am almost certain that it was known before Borcherds.|date=January 2018}}.

References

See chapter 14, theorem 14.6 of {{Apostol IANT}}
Peter J. Cameron, [https://www.amazon.co.jp/dp/0521457610/ Combinatorics: Topics, Techniques, Algorithms], (1994) Cambridge University Press, {{ISBN|0-521-45761-0}}
{{Citation | last1=Jacobi | first1=C. G. J. | title=Fundamenta nova theoriae functionum ellipticarum | url=https://archive.org/details/fundamentanovat00jacogoog | publisher=Borntraeger|place=Königsberg | language=Latin | isbn=978-1-108-05200-9 | id=Reprinted by Cambridge University Press 2012 | year=1829}}
{{Citation |last=Carlitz| first=L | title=A note on the Jacobi theta formula | url=https://projecteuclid.org/euclid.bams/1183524930| publisher=American Mathematical Society| year=1962}}
{{Citation |last=Wright| first=E. M.| title= An Enumerative Proof of An Identity of Jacobi| url=https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s1-40.1.55 | publisher=London Mathematical Society| year=1965}}

1. ^{{Cite journal|last=Andrews|first=George E.|date=1965-02-01|title=A simple proof of Jacobi’s triple product identity|url=https://www.ams.org/journals/proc/1965-016-02/S0002-9939-1965-0171725-X/|journal=Proceedings of the American Mathematical Society|language=en-US|volume=16|issue=2|pages=333–333|doi=10.1090/S0002-9939-1965-0171725-X|issn=0002-9939}}

External links

4 : Elliptic functions|Theta functions|Mathematical identities|Theorems in number theory

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