\[\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\pi^{1/2}x^{%
\nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)}\mathbf{F}%
\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1%
}{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac{\pi^{1/2}\left(x^{2}-1%
\right)^{\mu/2}}{2^{\nu+1}\Gamma\left(\mu-\nu\right)x^{\nu+\mu+1}}\mathbf{F}%
\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac%
{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),\]