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 $$ \frac{d^2}{dz^2}\theta_2(z,q)=-\frac{t^2}{(1+t^2)^2}+8\sum(-1)^n\frac{nq^{2n}(t^{2n}+t^{-2n})}{1-q^{2n}}; $$ 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?

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?