Closed form for the integral of a squared Legendre function

Question

Is there a closed form for the integral $$int_0^{pi/2}(P_nu^mu(costheta))^2,mathrm dtheta,quadmu>nugt-frac12$$ where $P_nu^mu(x)$ is the associated Legendre function of the first kind?

I encountered this integral while trying to derive explicit solutions for a certain Sturm-Liouville problem. I am primarily interested in $mu,nu$ being nonnegative integers, but a result that is valid for real $mu,nu$ (subject to the above restriction) is very much welcome.
Neither Maple nor Mathematica seem to be able to make a dent on this integral, but I was able to at least confirm that for $mu,nu$ an integer, I get results that are rational multiples of $pi$, which makes me believe there ought to be a (simple?) closed form, perhaps involving gamma functions.
I wasn't able to find anything in G&R or the DLMF that resembles this integral, so I am really stuck, and would appreciate any ideas on resolving this.

skbmoore · Answer

The best evaluation, a single sum, that I can derive is
$$ int_0^{pi/2} big( P_n^m(cos{theta}) big)^2 dtheta = frac{pi}{2} frac{(2m)!}{m!^4} Big( frac{(n+m)!}{(n-m)!} Big)^2 
{}_4F_3
 bigl( begin{smallmatrix} m+1/2, & m+1/2, & m-n, &m+n+1 \ m+1 & m+1 & 2m+1 end{smallmatrix} | 1bigr)  $$
I derived it from the answer given in Math Overflow 291481, which expresses the product of the associated Legendre polynomials as a single sum with powers of $sin^2{theta}$ within the summand.  The integration is then easy.  I then simplified the formula to that above.
It is doubtful that the generalized hypergeometric ${}_4F_3$ simplifies to a ratio of gamma functions.  I have a reference that has some ${}_4F_3$ evaluated at 1, but the 'numerator' parameters start off as $a, 1+a/2...$ and that's not the form of the answer.  Furthermore, I calculated it a few for small $m$ and $n,$ and if a ratio of gammas was in fact true, I wouldn't expect to get large primes in my answer.

Closed form for the integral of a squared Legendre function

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