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In mathematics, the Jacobi triple product is the mathematical identity: :\prod_^\infty \left( 1 - x^\right) \left( 1 + x^ y^2\right) \left( 1 +\frac\right) = \sum_^\infty x^ y^, for complex numbers ''x'' and ''y'', with , ''x'', < 1 and ''y'' ≠ 0. It was introduced by in his work ''
Fundamenta Nova Theoriae Functionum Ellipticarum ''Fundamenta nova theoriae functionum ellipticarum'' (New Foundations of the Theory of Elliptic Functions) is a book on Jacobi elliptic functions by Carl Gustav Jacob Jacobi.Given in Latin style as ''Carolo Gustavo Iacobo Iacobi'' in the book Th ...
''. The Jacobi triple product identity is the Macdonald identity for the affine root system of type ''A''1, 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 In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that :\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\righ ...
, which is itself a specific case of the Jacobi Triple Product Identity. Let x=q\sqrt q and y^2=-\sqrt. Then we have :\phi(q) = \prod_^\infty \left(1-q^m \right) = \sum_^\infty (-1)^n q^. The Jacobi Triple Product also allows the Jacobi
theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
to be written as an infinite product as follows: Let x=e^ and y=e^. Then the Jacobi theta function : \vartheta(z; \tau) = \sum_^\infty e^ can be written in the form :\sum_^\infty y^x^. Using the Jacobi Triple Product Identity we can then write the theta function as the product :\vartheta(z; \tau) = \prod_^\infty \left( 1 - e^\right) \left 1 + e^\right\left 1 + e^\right 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: :\sum_^\infty q^z^n = (q;q)_\infty \; \left(-\tfrac;q\right)_\infty \; (-zq;q)_\infty, where (a;q)_\infty is the infinite ''q''-Pochhammer symbol. It enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For , ab, <1 it can be written as :\sum_^\infty a^ \; b^ = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.


Proof

Let f_x(y) = \prod_^\infty \left( 1 - x^ \right)\left( 1 + x^ y^2\right)\left( 1 +x^y^\right) Substituting for and multiplying the new terms out gives :f_x(xy) = \fracf_x(y) = x^y^f_x(y) Since f_x is meromorphic for , y, > 0, it has a
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion ...
:f_x(y)=\sum_^\infty c_n(x)y^ which satisfies :\sum_^\infty c_n(x)x^ y^=x f_x(x y)=y^f_x(y)=\sum_^\infty c_(x)y^ so that :c_(x) = c_n(x)x^ = \dots = c_0(x) x^ and hence :f_x(y)=c_0(x) \sum_^\infty x^ y^


Evaluating

Showing that c_0(x) =1 is technical. One way is to set y= 1 and show both the numerator and the denominator of :\frac1 =\frac are weight 1/2 modular under z\mapsto -\frac, since they are also 1-periodic and bounded on the upper half plane the quotient has to be constant so that c_0(x)=c_0(0)=1.


Other proofs

A different proof is given by G. E. Andrews based on two identities of Euler. For the analytic case, see Apostol.Chapter 14, theorem 14.6 of


References

* Peter J. Cameron
''Combinatorics: Topics, Techniques, Algorithms''
(1994)
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
, * * *{{Citation , last=W 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 The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
, year=1965, doi=10.1112/jlms/s1-40.1.55 Elliptic functions Theta functions Mathematical identities Theorems in number theory Infinite products