In [an answer](https://mathoverflow.net/a/322394/41291) to my question [Enumeration of lattice paths of a specific type](https://mathoverflow.net/q/320457/41291) (which was in turn caused by an older one, ["Special" meanders](https://mathoverflow.net/q/146802/41291)) I encountered the series
$$
F(t,q):=\sum\_{n=1}^\infty(-1)^n\frac{q^nt}{\left(1-q^nt\right)^2}=-\sum\_{n=1}^\infty\frac{n(qt)^n}{1+q^n}=\sum\_{n=1}^\infty\left(\sum\_{d|n}(-1)^{\frac nd}dt^d\right)q^n.
$$
It resembles closely several famous things like parametrization of the Tate curve, or $q$-expansions of theta functions. In particular, $F(t,q)+F(t^{-1},q)$ is "almost" the derivative of the logarithmic derivative of $\theta\_2$, since
$$
\frac{d^2}{dz^2}\log\theta\_2(z,q)=-\frac{t^2}{(1+t^2)^2}+8\sum\_{n\geqslant1}(-1)^n\frac{nq^{2n}(t^{2n}+t^{-2n})}{1-q^{2n}}
$$
with $t=e^{iz}$; yet, I cannot go beyond that "almost" because of that silly plus sign in the denominator instead of minus.


What to make of this $F(t,q)$? Does it have some expression through known functions, or have some significance of its own?



"One half of a theta-function" - is there something in the literature about it?
Here is one (but likely not the simplest) way to evaluate $F(t,q)+F(t^{-1},q)$. Equation (2.19) of 


*Milne, Stephen C.*, [**Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions**](http://dx.doi.org/10.1023/A:1014865816981), Ramanujan J. 6, No. 1, 7-149 (2002). [arXiv:math/0008068](https://arxiv.org/pdf/math/0008068).


reads


$$
\frac{\mathrm{sn}(u,k)}{\mathrm{cn}(u,k)\mathrm{dn}(u,k)} = \frac{\pi}{2(1-k^2)K(k)}\left(\tan \frac{\pi u}{2K(k)} + 4\sum\_{n\geq 1} \frac{(-1)^n q^n}{1+q^n}\sin(n \pi u/K(k)) \right).
$$


Taking the $u$-derivative and setting $t = -e^{i\frac{\pi u}{K(k)}}$changes the sum (up to factors) into $\sum\_{n\geq 1} \frac{n q^n}{1+q^n} (t^{n}+t^{-n})$.
Rewriting the formula in terms of theta functions I believe this reasoning leads to


\begin{align}
F(t,q)+F(t^{-1},q) &= - \sum\_{n\geq 1} \frac{n q^n}{1+q^n}(t^n+t^{-n}) \\
&= \frac{1}{4\sin^2 y} + \frac{\theta\_2(0,q)^2\theta\_3(0,q)^2}{4} \left( \frac{\theta\_1(y,q)^2}{\theta\_4(y,q)^2}-\frac{\theta\_4(y,q)^2}{\theta\_1(y,q)^2}\right),
\end{align}
where $y$ and $t$ are related by $t= e^{2 i y}$.


