The Jacobi theta (or “thetanull”) function $\theta_3$ is defined by:
$$\theta_3(x)= \sum_{n \in \mathbb{Z}} \mathrm{e}^{-\pi n^2 x} = 1+ 2\sum_{n \in \mathbb{N}} \mathrm{e}^{-\pi n^2 x} \qquad \Re(x) > 0$$
In his 2018 paper, Dan Romik derived some very nice closed form expressions for the $d$-th derivative of $\theta_3(1)$. These expressions incorporate an intriguing integer sequence that might hide some undiscovered combinatoric structure.
Wondered whether similar closed forms could exist for subsequent integrals of $\theta_3$ from $0..1$, i.e.:
\begin{align} I^{(1)}(1)&=\int_0^1 \theta_3(x)\,dx \\ I^{(2)}(1)&=\int_0^1 \int_0^y \theta_3(x)\,dx\, dy\\ I^{(3)}(1)&=\int_0^1 \int_0^z \int_0^y \theta_3(x)\,dx\, dy\, dz\\ \\ \cdots \end{align}
Did manage to find a partial solution that is valid for any $d \in \mathbb{N}$ and $d=0$:
$$I^{(d)}(1)=2\left(\,\sum_{k=0}^{d+1}\frac{\left(-1\right)^{k+1} \zeta \! \left(2 k \right)}{\left(d -k \right)! \pi^{k}}+\left(-1\right)^{d} \sum_{n=1}^{\infty}\frac{{\mathrm e}^{-\pi \,n^{2}}}{\pi^{d} n^{2 d}}\right)$$
however, the second series didn't yield the desired closed form.
Found a few related questions about integrals of theta functions e.g. here and here, however when we take $q=\mathrm{e}^{-\pi}$, these turned out to be indefinite integrals "in disguise".
Do closed forms for $I^{d}(1)$ exist? If so, my prediction is that the inevitable factor $\Gamma\left(\frac14\right)$ will be part of it again :-)
