Ramanujan theta function

Short description: Mathematical 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

[math]\displaystyle{ f(a,b) = \sum_{n=-\infty}^\infty a^\frac{n(n+1)}{2} \; b^\frac{n(n-1)}{2} }[/math]

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

[math]\displaystyle{ f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty. }[/math]

Here, the expression [math]\displaystyle{ (a;q)_n }[/math] denotes the q-Pochhammer symbol. Identities that follow from this include

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \vartheta(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]\displaystyle{ 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 }[/math]

The special cases of Ramanujan's theta functions given by φ(q) := f(q, q) OEIS: A000122 and ψ(q) := f(q, q³) OEIS: A010054 also have the following integral representations:

[math]\displaystyle{ \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} }[/math]

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

[math]\displaystyle{ \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} }[/math]

and that

[math]\displaystyle{ \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} }[/math]

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

• Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. 32. Cambridge: Cambridge University Press. 
• Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. 96 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-83357-4. 
• Hazewinkel, Michiel, ed. (2001), "Ramanujan function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/r077200 
• Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press. ISBN 0-19-286189-1. 
• Weisstein, Eric W.. "Ramanujan Theta Functions". http://mathworld.wolfram.com/RamanujanThetaFunctions.html.