Some sums over irreducible polynomials

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form ${\sum }_{P}h(P)$ where the sum is over monic irreducible polynomials $P$ in ${\mathbb{F}}_q[T]$, the function $h$ is a rational function and the sum is considered in the $T^{-1}$-adic topology. As an example of our results, in ${\mathbb{F}}_2[T]$, the sum ${\sum }_P\frac{1}{P^k-1}$ always converges to a rational function, and is $0$ for $k=1$.