View source for Jacobi triple product

{{short description|Mathematical identity found by Jacobi in 1829}}
{{nofootnotes|date=May 2018}}

{{Redirects here|Triple product identity|the ternary operation on vector spaces|Triple product}}
In [[mathematics]], the '''Jacobi triple product''' is the identity:

:<math>\prod_{m=1}^\infty
\left( 1 - x^{2m}\right)
\left( 1 + x^{2m-1} y^2\right)
\left( 1 +\frac{x^{2m-1}}{y^2}\right)
= \sum_{n=-\infty}^\infty x^{n^2} y^{2n},
</math>
for complex numbers ''x'' and ''y'', with |''x''| &lt; 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 ''A''<sub>1</sub>, 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 <math>x=q\sqrt q</math> and <math>y^2=-\sqrt{q}</math>.  Then we have
:<math>\phi(q) = \prod_{m=1}^\infty \left(1-q^m \right) =
\sum_{n=-\infty}^\infty (-1)^n q^{\frac{3n^2-n}{2}}.</math>

The [[Rogers–Ramanujan identities]] follow with <math>x=q^2\sqrt q</math>, <math>y^2=-\sqrt{q}</math> and <math>x=q^2\sqrt q</math>, <math>y^2=-q\sqrt{q}</math>.

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

Let <math>x=e^{i\pi \tau}</math> and <math>y=e^{i\pi z}.</math>

Then the Jacobi theta function

:<math>
\vartheta(z; \tau) = \sum_{n=-\infty}^\infty e^{\pi {\rm{i}} n^2 \tau + 2 \pi {\rm{i}} n z}
</math>

can be written in the form

:<math>\sum_{n=-\infty}^\infty y^{2n}x^{n^2}. </math>

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

:<math>\vartheta(z; \tau) = \prod_{m=1}^\infty
\left( 1 - e^{2m \pi {\rm{i}} \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].
</math>

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 symbol|''q''-Pochhammer symbol]]s:

:<math>\sum_{n=-\infty}^\infty q^{\frac{n(n+1)}{2}}z^n =
(q;q)_\infty \; \left(-\tfrac{1}{z};q\right)_\infty \; (-zq;q)_\infty,</math>
where <math>(a;q)_\infty</math> is the infinite ''q''-Pochhammer symbol.

It enjoys a particularly elegant form when expressed in terms of the [[Ramanujan theta function]].  For <math>|ab|<1</math> it can be written as

:<math>\sum_{n=-\infty}^\infty a^{\frac{n(n+1)}{2}} \; b^{\frac{n(n-1)}{2}} = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.</math>

==Proof==

Let <math>f_x(y) = \prod_{m=1}^\infty \left( 1 - x^{2m} \right)\left( 1 + x^{2m-1} y^2\right)\left( 1 +x^{2m-1}y^{-2}\right)</math>

Substituting {{mvar|xy}} for {{mvar|y}} and multiplying the new terms out gives

:<math>f_x(xy) = \frac{1+x^{-1}y^{-2}}{1+xy^2}f_x(y) = x^{-1}y^{-2}f_x(y)</math>

Since <math>f_x</math> is meromorphic for <math>|y| > 0</math>, it has a [[Laurent series]]

:<math>f_x(y)=\sum_{n=-\infty}^\infty c_n(x)y^{2n}</math>

which satisfies 

:<math>\sum_{n=-\infty}^\infty c_n(x)x^{2n+1} y^{2n}=x f_x(x y)=y^{-2}f_x(y)=\sum_{n=-\infty}^\infty c_{n+1}(x)y^{2n}</math>

so that

:<math>c_{n+1}(x) = c_n(x)x^{2n+1} = \dots = c_0(x) x^{(n+1)^2}</math>

and hence

:<math>f_x(y)=c_0(x) \sum_{n=-\infty}^\infty x^{n^2} y^{2n}</math>

=== Evaluating {{math|''c''<sub>0</sub>(''x'')}} ===
Showing that <math>c_0(x) = 1</math> (the polynomial of ''x'' of <math>y^0</math> is ''1'') is technical. One way is to set <math>y= 1</math> and show both the numerator and the denominator of
:<math>\frac1{c_0(e^{2i\pi z})} =\frac{\sum\limits_{n=-\infty}^\infty e^{2i\pi n^2 z}}{\prod\limits_{m=1}^\infty (1-e^{2i\pi mz})(1+e^{2i\pi(2m-1)z})^2}</math>

are weight 1/2 [[modular form|modular]] under <math>z\mapsto -\frac{1}{4z}</math>, since they are also 1-periodic and bounded on the upper half plane the quotient has to be constant so that <math>c_0(x)=c_0(0)=1</math>.

=== Other proofs ===
A different proof is given by [[George Andrews (mathematician)|G. E. Andrews]] based on two identities of Euler.<ref>{{Cite journal|last=Andrews|first=George E.|date=1965-02-01|title=A simple proof of Jacobi's triple product identity|journal=Proceedings of the American Mathematical Society|language=en-US|volume=16|issue=2|pages=333–334|doi=10.1090/S0002-9939-1965-0171725-X|issn=0002-9939|doi-access=free}}</ref>

For the analytic case, see Apostol.<ref>Chapter 14, theorem 14.6 of {{Apostol IANT}}</ref> 

==References==
{{reflist}}
* 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=[[Leonard Carlitz|Carlitz]]| first=L | title=A note on the Jacobi theta formula | work=Bulletin of the American Mathematical Society | url=https://projecteuclid.org/euclid.bams/1183524930| publisher=[[American Mathematical Society]]| year=1962| volume=68 | issue=6 | pages=591–592 }}
*{{Citation |last=Wright| first=E. M.| title= An Enumerative Proof of An Identity of Jacobi| journal=Journal of the London Mathematical Society| pages=55–57| publisher=[[London Mathematical Society]]| year=1965| doi=10.1112/jlms/s1-40.1.55}}

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[[Category:Elliptic functions]]
[[Category:Theta functions]]
[[Category:Mathematical identities]]
[[Category:Theorems in number theory]]
[[Category:Infinite products]]