- Jacobi triple product
In
mathematics , the Jacobi triple product is a relation that re-expresses the Jacobitheta function , normally written as a series, as a product. This relationship generalizes other results, such as thepentagonal number theorem .Let "x" and "y" be complex numbers, with |"x"| < 1 and "y" not zero. Then:prod_{m=1}^infty left( 1 - x^{2m} ight)left( 1 + x^{2m-1} y^2 ight)left( 1 + x^{2m-1} y^{-2} ight)= sum_{n=-infty}^infty x^{n^2} y^{2n}.
This can easily be seen to be a relation on the Jacobi
theta function ; taking x=exp (ipi au) and y=exp(ipi z) one sees that the right hand side is:vartheta(z; au) = sum_{n=-infty}^infty exp (ipi n^2 au + 2i pi n z).
Euler's
pentagonal number theorem follows by taking x=q^{3/2} and y^2=-sqrt{q}. One then gets:phi(q) = prod_{m=1}^infty left(1-q^m ight) = sum_{n=-infty}^infty (-1)^n q^{(3n^2-n)/2}.,The Jacobi triple product enjoys a particularly elegant form when expressed in terms of the
Ramanujan theta function , which see. It also takes on a concise form when expressed in terms of q-Pochhammer symbols::sum_{n=-infty}^infty q^{n(n+1)/2}z^n = (q;q)_infty ; (-1/z;q)_infty ; (-zq;q)_infty.
Here, a;q)_infty is the infinite q-Pochhammer symbol.
Proof
This proof uses a simplified model of the
Dirac sea and follows the proof in Cameron (13.3) which is attributed toRichard 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:prod_{n>0}(1+q^{n-frac{1}{2z)(1+q^{n-frac{1}{2z^{-1})=left(sum_{linmathbb{Zq^{l^2/2}z^l ight)left(prod_{n>0}(1-q^n)^{-1} ight).
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 S is:sum{vcolon v > 0,vin S} - sum{vcolon v < 0, v otin S}
and the "particle number" of S is
:vcolon v>0,vin S}|-|{vcolon v<0,v otin S}|.
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 sum_{m,l} ?,q^mz^l for the number of states of energy m with l particles can be expressed as
:prod_{n>0}(1+q^{n-frac{1}{2z)(1+q^{n-frac{1}{2z^{-1}).
On the other hand, any state with l particles can be obtained from the lowest energy l-particle state, vcolon v
, by rearranging particles: take a partition lambda_1geqlambda_2geqcdotsgeqlambda_j of m' and move the top particle up by lambda_1 levels, the next highest particle up by lambda_2 levels, etc.... The resulting state has energy m'+frac{l^2}{2}, so the generating function can also be written as :left(sum_{linmathbb{Zq^{l^2/2}z^l ight)left(sum_{ngeq0}p(n)q^n ight)=left(sum_{linmathbb{Zq^{l^2/2}z^l ight)left(prod_{n>0}(1-q^n)^{-1} ight)
where p(n) is the
partition function . [http://arxiv.org/abs/math-ph/0309015 The uses of random partitions] byAndrei Okounkov contains a picture of a partition exciting the vacuum.References
* See chapter 14, theorem 14.6 of Apostol IANT
* Peter J. Cameron, "Combinatorics: Topics, Techniques, Algorithms", (1994) Cambridge University Press, ISBN 0-521-45761-0
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