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. Vol. 32. Cambridge: Cambridge University Press.
• Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. Vol. 96 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-83357-4.
• "Ramanujan function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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". MathWorld.