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

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

and

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

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 nome as:

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

References

• W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
• George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
• Weisstein, Eric W., "Ramanujan Theta Functions" from MathWorld.