# §14.11 Derivatives with Respect to Degree or Order

 14.11.1 $\frac{\partial}{\partial\nu}\mathsf{P}^{\mu}_{\nu}\left(x\right)=\pi\cot\left(% \nu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x\right)-\frac{1}{\pi}\mathsf{A}_{\nu% }^{\mu}(x),$
 14.11.2 $\frac{\partial}{\partial\nu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\tfrac{1}{2}% \pi^{2}\mathsf{P}^{\mu}_{\nu}\left(x\right)+\frac{\pi\sin\left(\mu\pi\right)}{% \sin\left(\nu\pi\right)\sin\left((\nu+\mu)\pi\right)}\mathsf{Q}^{\mu}_{\nu}% \left(x\right)-\tfrac{1}{2}\cot\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}% (x)+\tfrac{1}{2}\csc\left((\nu+\mu)\pi\right)\mathsf{A}_{\nu}^{\mu}(-x),$

where

 14.11.3 $\mathsf{A}_{\nu}^{\mu}(x)=\sin\left(\nu\pi\right)\left(\frac{1+x}{1-x}\right)^% {\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}% \Gamma\left(k-\nu\right)\Gamma\left(k+\nu+1\right)}{k!\Gamma\left(k-\mu+1% \right)}\*\left(\psi\left(k+\nu+1\right)-\psi\left(k-\nu\right)\right).$
 14.11.4 $\displaystyle\left.\frac{\partial}{\partial\mu}\mathsf{P}^{\mu}_{\nu}\left(x% \right)\right|_{\mu=0}$ $\displaystyle=\left(\psi\left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)% \mathsf{P}_{\nu}\left(x\right)+\mathsf{Q}_{\nu}\left(x\right),$ 14.11.5 $\displaystyle\left.\frac{\partial}{\partial\mu}\mathsf{Q}^{\mu}_{\nu}\left(x% \right)\right|_{\mu=0}$ $\displaystyle=-\tfrac{1}{4}\pi^{2}\mathsf{P}_{\nu}\left(x\right)+\left(\psi% \left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{Q}_{\nu}\left(x% \right).$

(14.11.1) holds if $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ is replaced by $P^{\mu}_{\nu}\left(x\right)$, provided that the factor $(\ifrac{(1+x)}{(1-x)})^{\mu/2}$ in (14.11.3) is replaced by $(\ifrac{(x+1)}{(x-1)})^{\mu/2}$. (14.11.4) holds if $\mathsf{P}^{\mu}_{\nu}\left(x\right)$, $\mathsf{P}_{\nu}\left(x\right)$, and $\mathsf{Q}_{\nu}\left(x\right)$ are replaced by $P^{\mu}_{\nu}\left(x\right)$, $P_{\nu}\left(x\right)$, and $Q_{\nu}\left(x\right)$, respectively.

See also Szmytkowski (2006, 2009, 2011, 2012), Cohl (2010, 2011) and Magnus et al. (1966, pp. 177–178).