§14.11 Derivatives with Respect to Degree or Order

14.11.1		$$\frac{\partial }{\partial \nu }{\mathsf{P}}_{\nu }^{\mu }\left(x\right)=\pi \cot \left(\nu \pi \right){\mathsf{P}}_{\nu }^{\mu }\left(x\right)-\frac{1}{\pi }{\mathsf{A}}_{\nu }^{\mu }(x),$$
ⓘ Symbols: : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${\mathsf{A}}_{\nu }^{\mu }(x)$ Referenced by: §14.11, §14.11 Permalink: http://dlmf.nist.gov/14.11.E1 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14

14.11.2		$$\frac{\partial }{\partial \nu }{\mathsf{Q}}_{\nu }^{\mu }\left(x\right)=\begin{array}{l}-\frac{1}{2}{\pi }^2{\mathsf{P}}_{\nu }^{\mu }\left(x\right)+\frac{\pi \sin \left(\mu \pi \right)}{\sin \left(\nu \pi \right)\sin \left((\nu +\mu )\pi \right)}{\mathsf{Q}}_{\nu }^{\mu }\left(x\right)-\frac{1}{2}\cot \left((\nu +\mu )\pi \right){\mathsf{A}}_{\nu }^{\mu }(x)\\ +\frac{1}{2}\csc \left((\nu +\mu )\pi \right){\mathsf{A}}_{\nu }^{\mu }(-x),\end{array}$$
ⓘ Symbols: : Ferrers function of the first kind, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${\mathsf{A}}_{\nu }^{\mu }(x)$ Referenced by: §14.11 Permalink: http://dlmf.nist.gov/14.11.E2 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14

where

14.11.3		$${\mathsf{A}}_{\nu }^{\mu }(x)=\sin \left(\nu \pi \right){\left(\frac{1+x}{1-x}\right)}^{\mu /2}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\frac{1}{2}x\right)}^k\mathrm{\Gamma }\left(k-\nu \right)\mathrm{\Gamma }\left(k+\nu +1\right)}{k!\mathrm{\Gamma }\left(k-\mu +1\right)}\left(\psi \left(k+\nu +1\right)-\psi \left(k-\nu \right)\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${\mathsf{A}}_{\nu }^{\mu }(x)$ Referenced by: §14.11 Permalink: http://dlmf.nist.gov/14.11.E3 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14

14.11.4		${{\displaystyle \frac{\partial }{\partial \mu }}{\mathsf{P}}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}$	$=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){\mathsf{P}}_{\nu }\left(x\right)+{\mathsf{Q}}_{\nu }\left(x\right),$
ⓘ Symbols: : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.11, §14.11 Permalink: http://dlmf.nist.gov/14.11.E4 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14
14.11.5		${{\displaystyle \frac{\partial }{\partial \mu }}{\mathsf{Q}}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}$	$=-\frac{1}{4}{\pi }^2{\mathsf{P}}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){\mathsf{Q}}_{\nu }\left(x\right).$
ⓘ Symbols: : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.11 Permalink: http://dlmf.nist.gov/14.11.E5 Encodings: TeX, pMML, png See also: Annotations for §14.11 and Ch.14

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