In mathematics, particularly -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa Ramanujan.
Definition
The Ramanujan theta function is defined as
:<math>f(a,b) = \sum_{n=-\infty}^\infty
a^\frac{n(n+1)}{2} \; b^\frac{n(n-1)}{2} </math>
for . The Jacobi triple product identity then takes the form
:<math>f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.</math>
Here, the expression <math>(a;q)_n</math> denotes the -Pochhammer symbol. Identities that follow from this include
:<math>\varphi(q) = f(q,q) = \sum_{n=-\infty}^\infty q^{n^2} =
{\left(-q;q^2\right)_\infty^2 \left(q^2;q^2\right)_\infty} </math>
and
:<math>\psi(q) = f\left(q,q^3\right) = \sum_{n=0}^\infty q^\frac{n(n+1)}{2} =
{\left(q^2;q^2\right)_\infty}{(-q; q)_\infty} </math>
and
:<math>f(-q) = f\left(-q,-q^2\right) = \sum_{n=-\infty}^\infty (-1)^n q^\frac{n(3n-1)}{2} =
(q;q)_\infty </math>
This last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
:<math>\vartheta_{00}(w, q)=f\left(qw^2,qw^{-2}\right)</math>
Integral representations
We have the following integral representation for the full two-parameter form of Ramanujan's theta function:
:<math>
\begin{align}
f(a,b) = 1 + \int_0^{\infty} \frac{2a e^{-\frac12 t^2{\sqrt{2\pi\left[
\frac{1 - a \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right)}{
a^3 b - 2a \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right) + 1}
\right] dt + \\
\int_0^{\infty} \frac{2b e^{-\frac12 t^2{\sqrt{2\pi\left[
\frac{1 - b \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right)}{
a b^3 - 2b \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right) + 1}
\right] dt
\end{align}
</math>
The special cases of Ramanujan's theta functions given by and also have the following integral representations: