Ramanujan theta function

In mathematics, particularly $q$-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


 * $$f(a,b) = \sum_{n=-\infty}^\infty

a^\frac{n(n+1)}{2} \; b^\frac{n(n-1)}{2} $$

for $|ab| < 1$. The Jacobi triple product identity then takes the form


 * $$f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.$$

Here, the expression $$(a;q)_n$$ denotes the $q$-Pochhammer symbol. Identities that follow from this include


 * $$\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} $$

and


 * $$\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} $$

and


 * $$f(-q) = f\left(-q,-q^2\right) = \sum_{n=-\infty}^\infty (-1)^n q^\frac{n(3n-1)}{2} =

(q;q)_\infty $$

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:


 * $$\vartheta(w, q)=f\left(qw^2,qw^{-2}\right)$$

Integral representations
We have the following integral representation for the full two-parameter form of Ramanujan's theta function:



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 $$

The special cases of Ramanujan's theta functions given by $φ(q) := f(q, q)$ and $ψ(q) := f(q, q^{3})$  also have the following integral representations:



\begin{align} \varphi(q) & = 1 + \int_0^{\infty} \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[\frac{4q \left(1-q^2 \cosh\left( \sqrt{2 \log q} \,t\right)\right)}{q^4-2 q^2 \cosh\left(\sqrt{2 \log q} \,t\right) + 1} \right] dt \\[6pt] \psi(q) & = \int_0^{\infty} \frac{2 e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[\frac{1-\sqrt{q} \cosh\left(\sqrt{\log q} \,t\right)}{q-2 \sqrt{q} \cosh\left(\sqrt{\log q} \,t\right) + 1} \right] dt \end{align} $$

This leads to several special case integrals for constants defined by these functions when $q := e^{−kπ}$ (cf. theta function explicit values). In particular, we have that



\begin{align} \varphi\left(e^{-k\pi}\right) & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^{k\pi} \left(e^{2k\pi} - \cos\left(\sqrt{2\pi k} \,t\right)     \right)}{e^{4k\pi} - 2 e^{2k\pi} \cos\left(\sqrt{2\pi k} \,t\right) + 1} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^\pi \left(e^{2\pi} - \cos\left(\sqrt{2\pi} \,t\right)     \right)}{e^{4\pi} - 2 e^{2\pi} \cos\left(\sqrt{2\pi} \,t\right) + 1} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{\sqrt{2 + \sqrt{2}}}{2} & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^{2\pi} \left(e^{4\pi} - \cos\left(2 \sqrt{\pi} \,t\right)     \right)}{e^{8\pi} - 2 e^{4\pi} \cos\left(2 \sqrt{\pi} \,t\right) + 1} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{\sqrt{1 + \sqrt{3}}}{2^\frac14 3^\frac38} & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^{3\pi} \left(e^{6\pi} - \cos\left(\sqrt{6 \pi} \,t\right)     \right)}{e^{12\pi} - 2 e^{6\pi} \cos\left(\sqrt{6 \pi} \,t\right) + 1} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{\sqrt{5 + 2 \sqrt{5}}}{5^\frac34} & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^{5\pi} \left(e^{10\pi} - \cos\left(\sqrt{10 \pi} \,t\right)     \right)}{e^{20\pi} - 2 e^{10\pi} \cos\left(\sqrt{10 \pi} \,t\right) + 1} \right] dt \end{align} $$

and that



\begin{align} \psi\left(e^{-k\pi}\right) & = \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{\cos\left(\sqrt{k \pi} \,t\right) - e^\frac{k\pi}{2}}{ \cos\left(\sqrt{k \pi} \,t\right) - \cosh\frac{k\pi}{2}} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{e^\frac{\pi}{8}}{2^\frac58} & = \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{\cos\left(\sqrt{\pi} \,t\right) - e^\frac{\pi}{2}}{ \cos\left(\sqrt{\pi} \,t\right) - \cosh\frac{\pi}{2}} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{e^\frac{\pi}{4}}{2^\frac54} & = \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{\cos\left(\sqrt{2 \pi} \,t\right) - e^\pi}{ \cos\left(\sqrt{2 \pi} \,t\right) - \cosh \pi} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{\sqrt[4]{1 + \sqrt{2}} \, e^\frac{\pi}{16}}{2^\frac{7}{16}} & = \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{\cos\left(\sqrt{\frac{\pi}{2}} \,t\right) - e^\frac{\pi}{4}}{ \cos\left(\sqrt{\frac{\pi}{2}} \,t\right) - \cosh\frac{\pi}{4}} \right] dt \end{align} $$

Application in string theory
The Ramanujan theta function is used to determine the critical dimensions in bosonic string theory, superstring theory and M-theory.