Ramanujan theta function

{{Distinguish|text=the mock theta functions discovered by Ramanujan}}

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 {{math|{{abs|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:

: $\vartheta_{00}(w, q)=f\left(qw^2,qw^{-2}\right)$

Integral representations

We have the following integral representation for the full two-parameter form of Ramanujan's theta function:{{cite journal|last1=Schmidt|first1=M. D.|title=Square series generating function transformations|journal=Journal of Inequalities and Special Functions|date=2017|volume=8|issue=2|arxiv=1609.02803|url=http://www.ilirias.com/jiasf/repository/docs/JIASF8-2-11.pdf}}

: $\begin{align}
f(a,b) = 1 + \int_0^{\infty} \frac{2a e^{-\frac12 t^2}}{\sqrt{2\pi}}\left[
\frac{1 - a \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right)}{
a^3 b - 2a \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right) + 1}
\right] dt + \\
\int_0^{\infty} \frac{2b e^{-\frac12 t^2}}{\sqrt{2\pi}}\left[
\frac{1 - b \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right)}{
a b^3 - 2b \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right) + 1}
\right] dt
\end{align}$

The special cases of Ramanujan's theta functions given by {{math|φ(q) :{{=}} f(q, q)}} {{oeis|id=A000122}} and {{math|ψ(q) :{{=}} f(q, q³)}} {{oeis|id=A010054}} {{cite web|last1=Weisstein|first1=Eric W.|title=Ramanujan Theta Functions|url=http://mathworld.wolfram.com/RamanujanThetaFunctions.html|website=MathWorld|accessdate=29 April 2018}} 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 {{math|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.

References

{{cite book |first=W. N. |last=Bailey |title=Generalized Hypergeometric Series |year=1935 |series=Cambridge Tracts in Mathematics and Mathematical Physics |volume=32 |publisher=Cambridge University Press |location=Cambridge }}
{{cite book |first1=George |last1=Gasper |first2=Mizan |last2=Rahman |title=Basic Hypergeometric Series |edition=2nd |year=2004 |series=Encyclopedia of Mathematics and Its Applications |volume=96 |publisher=Cambridge University Press |location=Cambridge |isbn=0-521-83357-4 }}
{{springer|title=Ramanujan function|id=p/r077200|mode=cs1}}
{{cite book |last=Kaku |first=Michio |year=1994 |title=Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension |location=Oxford |publisher=Oxford University Press |isbn=0-19-286189-1 }}
{{Mathworld|RamanujanThetaFunctions|Ramanujan Theta Functions}}

Category:Q-analogs

Category:Elliptic functions

Category:Theta functions

Category:Srinivasa Ramanujan