Ramanujan theta function explained

In mathematics, particularly -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 ^{inf ty a} _{n =-inf ty} ^{(n(n +1 ))/(2)} b ^{(n(n -1 ))/(2)}

for . 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 -Pochhammer symbol. Identities that follow from this include\varphi(q) = f(q,q) = \sum ^{inf ty} _{n =-inf ty} q ^{n ^{2}} = {\left( - q;q ^{2} _{inf ty} \left(q ^{2;q} ^{2 \right)} ^{inf ty}}

and

\psi(q) = f\left(q,q ^{3\right)} = \sum ^{inf ty} _{n =0} q ^{(n(n +1 ))/(2)} = {\left(q ^{2;q} ^{2 \right)} ^{inf ty}{( - q;} q) _{infty}}

and

f( - q) = f\left( - q, - q ^{2\right)} = \sum ^{inf ty} _{n =-inf ty} ( -1 ) ^{n} q ^{(n( 3 n -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:^[1]

f(a,b) = 1 + \int ^{inf ty} _{0} (2 a e ^{- (12)/(^{2} t)})/(\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^ \frac\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 and ^[2] also have the following integral representations:

\begin{align} \varphi(q) & = 1 + \int ^{inf ty} _{0} (e ^{- (12)/(^{2} t)})/(\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^ \frac \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

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 ^{inf ty} _{0} (e ^{- (12)/(^{2} t)})/(\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 & = 1 + \int_0^\infty \frac \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 \cdot \frac & = 1 + \int_0^\infty \frac \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 \cdot \frac & = 1 + \int_0^\infty \frac \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 \cdot \frac & = 1 + \int_0^\infty \frac \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

and that

\begin{align} \psi\left(e ^{-k\pi} \right) & = \int ^{inf ty} _{0} (e ^{- (12)/(^{2} t)})/(\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 \cdot \frac & = \int_0^\infty \frac \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 \cdot \frac & = \int_0^\infty \frac \left[\frac{\cos\left(\sqrt{2 \pi} \,t\right) - e^\pi}{ \cos\left(\sqrt{2 \pi} \,t\right) - \cosh \pi} \right] dt \\[6pt] \frac \cdot \frac & = \int_0^\infty \frac \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

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

• Book: Bailey, W. N. . Generalized Hypergeometric Series . 1935 . Cambridge Tracts in Mathematics and Mathematical Physics . 32 . Cambridge University Press . Cambridge .
• Book: George . Gasper . Mizan . Rahman . Basic Hypergeometric Series . 2nd . 2004 . Encyclopedia of Mathematics and Its Applications . 96 . Cambridge University Press . Cambridge . 0-521-83357-4 .
  • Book: Kaku, Michio . 1994 . Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension . Oxford . Oxford University Press . 0-19-286189-1 .

Notes and References

1. Schmidt. M. D.. Square series generating function transformations. Journal of Inequalities and Special Functions. 2017. 8. 2. 1609.02803.
2. Web site: Weisstein. Eric W.. Ramanujan Theta Functions. MathWorld. 29 April 2018.