\[P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\frac{1}{2}%
\right)(\sinh\xi)^{m}}{2^{m}\pi^{1/2}\Gamma\left(n-m+\frac{1}{2}\right)\Gamma%
\left(m+\frac{1}{2}\right)}\*\int_{0}^{\pi}\frac{(\sin\phi)^{2m}}{(\cosh\xi+%
\cos\phi\sinh\xi)^{n+m+(1/2)}}\,\mathrm{d}\phi,\]