# 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, 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, ISBN 978-1-55608-010-4
• 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.