\[P^{\mu}_{\nu+1}\left(x\right)Q^{\mu}_{\nu}\left(x\right)-P^{\mu}_{\nu}\left(x%
\right)Q^{\mu}_{\nu+1}\left(x\right)=e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1%
\right)}{\Gamma\left(\nu-\mu+2\right)}.\]