"The" theta function is an ambiguous concept, but one definition I have found is:
$$ \theta(z;q) = (z;q)_\infty(q/z;q)_\infty = \frac{1}{(q;q)_\infty}\sum_{k \in \mathbb{Z}}z^k q^{\binom{k}{2}} \tag{$\ast$} $$
I found this recently on arXiv but maybe there is a textbook. The second equation is known as the Jacobi triple product:
$$ (q;q)_\infty(z;q)_\infty(q/z;q)_\infty = \sum_{k \in \mathbb{Z}}z^k q^{\binom{k}{2}} $$
Therefore it might make better sense to call the theta function this result of this triple product rather than definition ($\ast$) Before I forget $(z;q)$ is an abbreviation for a factorial called the Pochammer symbol
$$ (z;q) = \prod_{i=0}^\infty (1 - z q^i)$$
My question is if there is an easy expansion for the reciprocal of the theta function. I certainly could not think of one:
$$ \frac{1}{\theta(z;q)} = \frac{1}{(z;q)_\infty} \frac{1}{(q/z;q)_\infty} = (q;q)_\infty \frac{1}{\sum_{k \in \mathbb{Z}}z^k q^{\binom{k}{2}}} \tag{$\ast$} = \sum \;?\,?\,?$$