Ramanujan theta function

This is not about 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 Srinivasa Ramanujan.

Definition

The Ramanujan theta function is defined as

$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$

for |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

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

and

$f(q,q^3) = \sum_{n=0}^\infty q^{n(n+1)/2} = {(q^2;q^2)_\infty}{(-q; q)_\infty}$

and

$f(-q,-q^2) = \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)/2} = (q;q)_\infty$

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:

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

Application in string theory

The Ramanujan theta function is used to determine the critical dimensions in Bosonic string theory, SuperG-string Theory and S&M-theory.

References

• Weisstein, Eric W., "Ramanujan Theta Functions", MathWorld.