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| In [[mathematics]], the '''Jacobi triple product''' is the mathematical identity:
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| :<math>\prod_{m=1}^\infty
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| \left( 1 - x^{2m}\right)
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| \left( 1 + x^{2m-1} y^2\right)
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| \left( 1 + x^{2m-1} y^{-2}\right)
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| = \sum_{n=-\infty}^\infty x^{n^2} y^{2n},
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| </math>
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| for complex numbers ''x'' and ''y'', with |''x''| < 1 and ''y'' ≠ 0.
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| It was introduced by {{harvs|txt|authorlink=Carl Gustav Jacob Jacobi|last=Jacobi|year=1829}} in his work ''[[Fundamenta Nova Theoriae Functionum Ellipticarum]]''.
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| 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]].
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| == Properties ==
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| 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.
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| Let <math>x=q^{3/2}</math> and <math>y^2=-\sqrt{q}</math>. Then we have
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| :<math>\phi(q) = \prod_{m=1}^\infty \left(1-q^m \right) =
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| \sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.\, </math>
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| The Jacobi Triple Product also allows the Jacobi [[theta function]] to be written as an infinite product as follows:
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| Let <math>x=e^{i\pi \tau}</math> and <math>y=e^{i\pi z}.</math>
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| Then the Jacobi theta function
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| :<math>
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| \vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)
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| </math>
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| can be written in the form
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| :<math>\sum_{n=-\infty}^\infty y^{2n}x^{n^2}. </math>
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| Using the Jacobi Triple Product Identity we can then write the theta function as the product
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| :<math>\vartheta(z; \tau) = \prod_{m=1}^\infty
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| \left( 1 - \exp(2m \pi i \tau)\right)
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| \left( 1 + \exp((2m-1) \pi i \tau + 2 \pi i z)\right)
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| \left( 1 + \exp((2m-1) \pi i \tau -2 \pi i z)\right).
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| </math>
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| 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]]s:
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| :<math>\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n =
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| (q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.</math>
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| Where <math>(a;q)_\infty</math> is the infinite ''q''-Pochhammer symbol.
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| 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
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| :<math>\sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2} = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.</math>
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| ==Proof==
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| This proof uses a simplified model of the [[Dirac sea]] and follows the proof in Cameron (13.3) which is attributed to [[Richard Borcherds]]. It treats the case where the power series are formal. For the analytic case, see Apostol. The Jacobi triple product identity can be expressed as
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| :<math>\prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})=\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\prod_{n>0}(1-q^n)^{-1}\right).</math>
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| A ''level'' is a [[half-integer]]. The vacuum state is the set of all negative levels. A state is a set of levels whose symmetric difference with the vacuum state is finite. The ''energy'' of the state <math>S</math> is
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| :<math>\sum\{v\colon v > 0,v\in S\} - \sum\{v\colon v < 0, v\not\in S\}</math>
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| and the ''particle number'' of <math>S</math> is
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| :<math>|\{v\colon v>0,v\in S\}|-|\{v\colon v<0,v\not\in S\}|.</math>
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| An unordered choice of the presence of finitely many positive levels and the absence of finitely many negative levels (relative to the vacuum) corresponds to a state, so the generating function <math>\textstyle\sum_{m,l} s(m,l)q^mz^l</math> for the number <math>s(m,l)</math> of states of energy <math>m</math> with <math>l</math> particles can be expressed as
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| :<math>\prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1}).</math>
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| On the other hand, any state with <math>l</math> particles can be obtained from the lowest energy <math>l-</math>particle state, <math>\{v\colon v<l\}</math>, by rearranging particles: take a partition <math>\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_j</math> of <math>m'</math> and move the top particle up by <math>\lambda_1</math> levels, the next highest particle up by <math>\lambda_2</math> levels, etc.... The resulting state has energy <math>m'+\frac{l^2}{2}</math>, so the generating function can also be written as
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| :<math>\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\sum_{n\geq0}p(n)q^n\right)=\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\prod_{n>0}(1-q^n)^{-1}\right)</math>
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| where <math>p(n)</math> is the [[Partition function (number theory)|partition function]]. [http://arxiv.org/abs/math-ph/0309015 The uses of random partitions] by [[Andrei Okounkov]] contains a picture of a partition exciting the vacuum.
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| == Notes ==
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| <references/>
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| ==References==
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| * See chapter 14, theorem 14.6 of {{Apostol IANT}}
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| * Peter J. Cameron, ''Combinatorics: Topics, Techniques, Algorithms'', (1994) Cambridge University Press, ISBN 0-521-45761-0
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| *{{Citation | last1=Jacobi | first1=C. G. J. | title=Fundamenta nova theoriae functionum ellipticarum | url=http://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}}
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| [[Category:Elliptic functions]]
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| [[Category:Theta functions]]
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| [[Category:Mathematical identities]]
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| [[Category:Number theory]]
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Hello. Let me introduce the author. Her title is Emilia Shroyer but it's not the most feminine name out there. I am a meter reader but I plan on changing it. Her family members lives in Minnesota. To perform baseball is the hobby he will by no means quit performing.
My web-site; over the counter std test (continue reading this..)