Ramanujan theta function

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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.


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}


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


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

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[edit]

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