I'm looking to compute

$${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr 1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$

for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if there is any reduction that can be applied, such as Thomae's theorem, which is mentioned in this post (though I don't think I can apply it here).

I would also be interested in any reductions, or if this evaluation was $0$ or had a definite sign like $(-1)^m$. Apologies in advance as I am a geometric analyst looking to compute things with eigenfunctions of the laplacian on the hyperbolic disk - any references there would also be useful!