= Solution
Expand both generating functions and use <orthogonality> of the <Legendre polynomials>, with $\int_{-1}^1P_nP_m=2\delta_{nm}/(2n+1)$. The allowed exchanges give
$$
\int_{-1}^1\frac{dt}{\sqrt{1-2wt+w^2}\sqrt{1-2zt+z^2}}=2\sum_{n\geq0}\frac{(wz)^n}{2n+1}.
$$
Hence it depends only on $r=wz$. Explicitly,
$$
\boxed{F(r)=2\int_0^1\frac{ds}{1-rs^2}=\begin{cases}2\operatorname{artanh}(\sqrt r)/\sqrt r,&r>0,\\2,&r=0,\\2\arctan(\sqrt{-r})/\sqrt{-r},&r<0.\end{cases}}
$$
The <series> form alone already proves the required dependence.
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