"One half of a theta-function" - is there something in the literature about it?

Question

In an answer to my question Enumeration of lattice paths of a specific type (which was in turn caused by an older one, "Special" meanders) 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?

Answer 1

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, Ramanujan J. 6, No. 1, 7-149 (2002). arXiv: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 $t= e^{2 i y}$.