I am considering solutions to the generalized Legendre differential equation, e.g. the first kind, $P_{\nu}^{\mu}(x)$, on the interval $(-1, 1)$. Then I am wondering, for this function to be square integrable with respect to the meassure $dx\frac{1}{1-x^2}$, is it necessary that both $\nu$ and $\mu$ are integers?
My guess, based on some examples, is that the two parameters do not have to be integers for the previously stated condition and what is necessary is that their difference is an integer. But I can't prove this and further can not specify the range of the parameters, e.g. it also seems that $|\mu|\leq|\nu|$.
Can anybody offer a helping hand on this issue? A sketchy proof or some reference book with a proof will be much appreciated.
In the book by Magnus, Oberhettinger and Soni, Formulas and theorems for the special functions of mathematical physics, one can find "proof" (a collection of evidences) that when both $\mu$ and $\nu$ are real numbers, then as long as $|\nu|-|\mu|$ is a non-negative integer, the solution $P_{\nu}^{-|\mu|}(x)$ is square integrable on the domain (-1, 1) with respect to the defined measure.