Consider the analytic space $\mathbb{C}^{*}$ with coordinate $z$. Let $q$ be some parameter with $|q|<1$ and define the analytic function $$\theta(z;q):=\sum_{n\in\mathbb{Z}}q^{\binom{n}{2}}(-z)^{n}.$$ *Remark* that this is I think usually denoted $\theta_{11}$, and is the theta function corresponding to the trivial two torsion point on the elliptic curve $E_{q}=\mathbb{C}^{*}/q^{\mathbb{Z}}$.

**Question.** Is there an analytic function on $\mathbb{C}^{*}$, denoted $s(z)$, with the property $$s(qz)-s(z)=\theta(z).$$ Moreover can it be expressed neatly as some $q$-series? What I would *really* like is a reference to a text where such identities are collected in bulk so I don´t have to spend too much time trying to invent them myself.