nLab
Jacobi triple product
Statetement
The equation
∏ n = 1 ∞ ( 1 − x 2 n ) ( 1 − x 2 n − 1 y 2 ) ( 1 − x 2 n − 1 y − 2 ) = ∑ n = − ∞ ∞ x 2 n y 2 n
\underoverset{n=1}{\infty}{\prod}
\left(
1- x^{2n}
\right)
\left(
1- x^{2n-1} y^2
\right)
\left(
1- x^{2n-1} y^{-2}
\right)
=
\underoverset{n = -\infty}{\infty}{\sum}
x^{2n} y^{2n}
Equivalently this is a relation between the four Jacobi theta functions (see there and at Jacobi form
References
Due to
Carl Jacobi , Fundamenta nova theoriae functionum ellipticarum (in Latin), Königsberg: Borntraeger, ISBN 978-1-108-05200-9, Reprinted by Cambridge University Press 2012
Review includes
A large collection of identities between the various Jacobi theta functions related is at
Created on September 10, 2014 at 19:41:18.
See the history of this page for a list of all contributions to it.