Ramanujan theta function

In mathematics, particularly [[q-analog|Template:Mvar-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

$f(a,b)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}^{\frac{n(n+1)}{2}}{b}^{\frac{n(n-1)}{2}}$

for Template:Math. The Jacobi triple product identity then takes the form

$f(a,b)=(-a;ab{)}_{\mathrm{\infty }}(-b;ab{)}_{\mathrm{\infty }}(ab;ab{)}_{\mathrm{\infty }}.$

Here, the expression $(a;q{)}_n$ denotes the [[q-Pochhammer symbol|Template:Mvar-Pochhammer symbol]]. Identities that follow from this include

$\phi (q)=f(q,q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n^2}={(-q;q^2)}_{\mathrm{\infty }}^2{(q^2;q^2)}_{\mathrm{\infty }}$

and

$\psi (q)=f(q,q^3)=\sum _{n=0}^{\mathrm{\infty }}{q}^{\frac{n(n+1)}{2}}={(q^2;q^2)}_{\mathrm{\infty }}(-q;q{)}_{\mathrm{\infty }}$

and

$f(-q)=f(-q,-q^2)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{\frac{n(3n-1)}{2}}=(q;q{)}_{\mathrm{\infty }}$

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:

${\vartheta }_{00}(w,q)=f(q{w}^2,q{w}^{-2})$

Integral representations

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

$f(a,b)=1+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{2a{e}^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{1-a\sqrt{ab}\cosh \left(\sqrt{\log ab}t\right)}{a^{3}b-2a\sqrt{ab}\cosh \left(\sqrt{\log ab}t\right)+1}\right]dt+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{2b{e}^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{1-b\sqrt{ab}\cosh \left(\sqrt{\log ab}t\right)}{a{b}^3-2b\sqrt{ab}\cosh \left(\sqrt{\log ab}t\right)+1}\right]dt$

The special cases of Ramanujan's theta functions given by Template:Math Template:Oeis and Template:Math Template:Oeis also have the following integral representations:

$\begin{array}{cc}\phi (q)& =1+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{4q(1-q^2\cosh \left(\sqrt{2\log q}t\right))}{q^4-2q^2\cosh \left(\sqrt{2\log q}t\right)+1}\right]dt\\ \psi (q)& =\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{2e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{1-\sqrt{q}\cosh \left(\sqrt{\log q}t\right)}{q-2\sqrt{q}\cosh \left(\sqrt{\log q}t\right)+1}\right]dt\end{array}$

This leads to several special case integrals for constants defined by these functions when Template:Math (cf. theta function explicit values). In particular, we have that

$\begin{array}{cc}\phi \left(e^{-k\pi }\right)& =1+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{4e^{k\pi }(e^{2k\pi }-\cos \left(\sqrt{2\pi k}t\right))}{e^{4k\pi }-2e^{2k\pi }\cos \left(\sqrt{2\pi k}t\right)+1}\right]dt\\ \frac{{\pi }^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}& =1+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{4e^{\pi }(e^{2\pi }-\cos \left(\sqrt{2\pi }t\right))}{e^{4\pi }-2e^{2\pi }\cos \left(\sqrt{2\pi }t\right)+1}\right]dt\\ \frac{{\pi }^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}\cdot \frac{\sqrt{2+\sqrt{2}}}{2}& =1+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{4e^{2\pi }(e^{4\pi }-\cos \left(2\sqrt{\pi }t\right))}{e^{8\pi }-2e^{4\pi }\cos \left(2\sqrt{\pi }t\right)+1}\right]dt\\ \frac{{\pi }^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}\cdot \frac{\sqrt{1+\sqrt{3}}}{2^{\frac{1}{4}}{3}^{\frac{3}{8}}}& =1+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{4e^{3\pi }(e^{6\pi }-\cos \left(\sqrt{6\pi }t\right))}{e^{12\pi }-2e^{6\pi }\cos \left(\sqrt{6\pi }t\right)+1}\right]dt\\ \frac{{\pi }^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}\cdot \frac{\sqrt{5+2\sqrt{5}}}{5^{\frac{3}{4}}}& =1+\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{4e^{5\pi }(e^{10\pi }-\cos \left(\sqrt{10\pi }t\right))}{e^{20\pi }-2e^{10\pi }\cos \left(\sqrt{10\pi }t\right)+1}\right]dt\end{array}$

and that

$\begin{array}{cc}\psi \left(e^{-k\pi }\right)& =\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{\cos \left(\sqrt{k\pi }t\right)-e^{\frac{k\pi }{2}}}{\cos \left(\sqrt{k\pi }t\right)-\cosh \frac{k\pi }{2}}\right]dt\\ \frac{{\pi }^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}\cdot \frac{e^{\frac{\pi }{8}}}{2^{\frac{5}{8}}}& =\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{\cos \left(\sqrt{\pi }t\right)-e^{\frac{\pi }{2}}}{\cos \left(\sqrt{\pi }t\right)-\cosh \frac{\pi }{2}}\right]dt\\ \frac{{\pi }^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}\cdot \frac{e^{\frac{\pi }{4}}}{2^{\frac{5}{4}}}& =\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{\cos \left(\sqrt{2\pi }t\right)-e^{\pi }}{\cos \left(\sqrt{2\pi }t\right)-\cosh \pi }\right]dt\\ \frac{{\pi }^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}\cdot \frac{\sqrt[4]{1+\sqrt{2}}{e}^{\frac{\pi }{16}}}{2^{\frac{7}{16}}}& =\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{1}{2}{t}^2}}{\sqrt{2\pi }}\left[\frac{\cos \left(\sqrt{\frac{\pi }{2}}t\right)-e^{\frac{\pi }{4}}}{\cos \left(\sqrt{\frac{\pi }{2}}t\right)-\cosh \frac{\pi }{4}}\right]dt\end{array}$

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

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