# §14.11 Derivatives with Respect to Degree or Order

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

where

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

(14.11.1) holds if $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ is replaced by $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$, 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 $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$, $\mathop{\mathsf{P}_{\nu}\/}\nolimits\!\left(x\right)$, and $\mathop{\mathsf{Q}_{\nu}\/}\nolimits\!\left(x\right)$ are replaced by $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$, $\mathop{P_{\nu}\/}\nolimits\!\left(x\right)$, and $\mathop{Q_{\nu}\/}\nolimits\!\left(x\right)$, respectively.

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