Ramanujan Theta Function - Definition

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

Here, the expression 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:

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