Ramanujan theta function

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 {{math|

: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_{00}(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:

: \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}

The special cases of Ramanujan's theta functions given by and 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 (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.

References

References

1. (2017). "Square series generating function transformations". Journal of Inequalities and Special Functions.
2. "Ramanujan Theta Functions".