Properties of $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$? I'm looking for references concerning the properties of the function $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$ where $\theta_{1}(z|\tau)$ is a Jacobi theta function defined here. I am trying to find special values of $z$ where I can rewrite this function in terms of modular forms (or maybe $E_2$) and other theta constants, or anything else that could appear there.
 A: Notation: I write
$$\begin{aligned}
    q_n &= \exp\frac{2\pi\mathrm{i}\tau}{n}
&   q &= q_1
&   w &= \exp(\mathrm{i}z)
\\  \dot{(\ )}
    &= \frac{1}{2\pi\mathrm{i}}\left.\frac{\partial(\ )}{\partial\tau}\right|_z
    = \frac{q_n}{n}\left.\frac{\partial(\ )}{\partial q_n}\right|_w
&&& (\ )' &= \left.\frac{\partial(\ )}{\partial z}\right|_\tau
    = \mathrm{i} w \left.\frac{\partial(\ )}{\partial w}\right|_{q_n}
\end{aligned}$$
Note that the $q$ used in the DLMF pages
is written $q_2$ in this answer; their $q^2$ simplifies to $q$ here.
Expressions with $w$ and/or $q_n$ are implicitly regarded as functions of $z$
and/or $\tau$, respectively.
Answer: Given
$$\begin{aligned}
    \theta_1(z|\tau) &= \sum_{k\in\mathbb{Z}} q_8^{(2k+1)^2} (-\mathrm{i}w)^{2k+1}
    = 2q_8\sum_{n=0}^\infty (-1)^n q^{n(n+1)/2}\sin\left((2n+1)z\right)
\\  &= -\mathrm{i} w q_8\prod_{n=1}^\infty
    (1 - w^2 q^n)(1 - w^{-2} q^{n-1})(1 - q^n)
\\  &= 2q_8\sin(z)\prod_{n=1}^\infty (1 - w^2 q^n)(1 - w^{-2} q^n)(1 - q^n)
\end{aligned}$$
we can use the heat equation
$$\theta_1''(z|\tau) = -8\dot{\theta}_1(z|\tau)$$
and thus arrive at
$$\begin{aligned}
    \frac{\theta_1''(z|\tau)}{\theta_1(z|\tau)}
    &= -8\frac{\dot{\theta}_1(z|\tau)}{\theta_1(z|\tau)}
    = -8\left(\ln\theta_1(z|\tau)\right)^\cdot
\\  &= -8q\left.\frac{\partial}{\partial q}
    \ln\left(2q_8\sin(z)\prod_{n=1}^\infty
    (1 - w^2 q^n)(1 - w^{-2} q^n)(1 - q^n)\right)\right|_w
\\  &= -1 + 8\left(\sum_{n=1}^\infty\frac{n\,w^2 q^n}{1 - w^2 q^n}
    + \sum_{n=1}^\infty\frac{n\,w^{-2} q^n}{1 - w^{-2} q^n}
    + \sum_{n=1}^\infty\frac{n\,q^n}{1 - q^n}\right)
\end{aligned}$$
In order to compress that expression, we apply the usual trick of
expanding the fractions to series, swapping the series nesting orders,
and simplifying the new inner series. Thus, for $|w^2 q| < 1$,
$$\sum_{n=1}^\infty\frac{n\,w^2 q^n}{1 - w^2 q^n}
    = \sum_{n=1}^\infty n\sum_{m=1}^\infty w^{2m} q^{nm}
    = \sum_{m=1}^\infty w^{2m}\sum_{n=1}^\infty n\,q^{nm}
    = \sum_{m=1}^\infty w^{2m}\frac{q^m}{(1 - q^m)^2}$$
Doing the same with the other series, we obtain
$$\frac{\theta_1''(z|\tau)}{\theta_1(z|\tau)}
= -1 + 8\sum_{m=1}^\infty \frac{q^m}{(1 - q^m)^2}\left(1 + 2\cos(2mz)\right)
\qquad (|\Im z| < \pi\,\Im\tau)$$
Note that the right-hand side has no singularity at $z=0$ and yields
$$\lim_{z\to 0}\frac{\theta_1''(z|\tau)}{\theta_1(z|\tau)} =
-1 + 24\sum_{m=1}^\infty \frac{q^m}{(1 - q^m)^2} = -\mathrm{E}_2(\tau)$$
which is quasimodular in $\tau$.
