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.



Analytic function with q- difference equation involving theta
No, because the constant coefficient $c\_0=q^0=1$ of your Laurent series is non-zero. Look at
$$f(z) =c\_0\frac{\log z}{\log q}+ \sum\_{n\ne 0} \frac{q^{n(n-1)/2}}{q^n-1} (-z)^n, \qquad f(qz)-f(z)=\theta(z)\bmod \frac{c\_0 2i\pi }{\log q}$$ **Assuming your $s(z)$ exists** let
$$g(w)=s(e^w)-f(e^w)$$
which is entire, $\log q$ periodic, and $$g(w+2i\pi)= g(w)-c\_0\frac{2i\pi}{\log q} $$
Therefore, $\frac{g(w)-g(0)}{w}$ is entire and bounded so that $g(w)=g(0)+wg'(0)$.


$g(w+2i\pi)= g(w)-c\_0\frac{2i\pi}{\log q}$ gives that $g'(0)\ne 0$, contradicting that $g$ is $\log q$-periodic, contradicting that $s(z)$ is analytic on $\Bbb{C}^\*$.


