How to prove this relation?

Question

Is the relation $$\lim_{x\rightarrow 1}\frac{Q_n^m(x)}{P_n^m(x)}=\frac{\pi}{2}\cot m\pi$$ correct? Here P and Q are the associated Legendre polynomials of the first and second kind respectively. Does anybody know how to prove it, or some references that I can refer to? How about the limit at $x\rightarrow -1$? Is there a relation $$\lim_{x\rightarrow -1}\frac{Q_n^m(x)}{P_n^m(x)}=\frac{\pi}{2}\cot n\pi~?$$

Answer 1

Using the definitions of the Ferrers functions here, we have

$$\frac{Q_\nu^\mu(z)}{P_\nu^\mu(z)}=\frac{\pi}{2}\left(\cot(\pi\mu )-\csc(\pi\mu)\frac{\Gamma(\nu +\mu +1)}{\Gamma (\nu -\mu +1)}\frac{\Gamma(1-\mu)}{\Gamma(1+\mu)}\left(\frac{1-z}{1+z}\right)^{\mu }\frac{_2F_1\left({{-\nu,1+\nu}\atop{1+\mu}}\mid\frac{1-z}{2}\right)}{_2F_1\left({{-\nu ,1+\nu}\atop{1-\mu}}\mid\frac{1-z}{2}\right)}\right)$$

Letting $z=1$ proves the identity in the OP.