§14.10 Recurrence Relations and Derivatives

14.10.1		$${\mathsf{P}}_{\nu }^{\mu +2}\left(x\right)+2(\mu +1)x{\left(1-x^2\right)}^{-1/2}{\mathsf{P}}_{\nu }^{\mu +1}\left(x\right)+(\nu -\mu )(\nu +\mu +1){\mathsf{P}}_{\nu }^{\mu }\left(x\right)=0,$$
ⓘ Symbols: : Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.10 Permalink: http://dlmf.nist.gov/14.10.E1 Encodings: TeX, pMML, png See also: Annotations for §14.10 and Ch.14

14.10.2		$${\left(1-x^2\right)}^{1/2}{\mathsf{P}}_{\nu }^{\mu +1}\left(x\right)-(\nu -\mu +1){\mathsf{P}}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1)x{\mathsf{P}}_{\nu }^{\mu }\left(x\right)=0,$$
ⓘ Symbols: : Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.10.E2 Encodings: TeX, pMML, png See also: Annotations for §14.10 and Ch.14

14.10.3		$$(\nu -\mu +2){\mathsf{P}}_{\nu +2}^{\mu }\left(x\right)-(2\nu +3)x{\mathsf{P}}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1){\mathsf{P}}_{\nu }^{\mu }\left(x\right)=0,$$
ⓘ Symbols: : Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.5.3 (modified) Referenced by: §14.10, §14.14, §14.21(iii) Permalink: http://dlmf.nist.gov/14.10.E3 Encodings: TeX, pMML, png See also: Annotations for §14.10 and Ch.14

14.10.4		$$\left(1-x^2\right)\frac{d{\mathsf{P}}_{\nu }^{\mu }\left(x\right)}{dx}=(\mu -\nu -1){\mathsf{P}}_{\nu +1}^{\mu }\left(x\right)+(\nu +1)x{\mathsf{P}}_{\nu }^{\mu }\left(x\right),$$
ⓘ Symbols: : Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.5(i) Permalink: http://dlmf.nist.gov/14.10.E4 Encodings: TeX, pMML, png See also: Annotations for §14.10 and Ch.14

14.10.5		$$\left(1-x^2\right)\frac{d{\mathsf{P}}_{\nu }^{\mu }\left(x\right)}{dx}=(\nu +\mu ){\mathsf{P}}_{\nu -1}^{\mu }\left(x\right)-\nu x{\mathsf{P}}_{\nu }^{\mu }\left(x\right).$$
ⓘ Symbols: : Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.5.4 (modified) Referenced by: §14.10 Permalink: http://dlmf.nist.gov/14.10.E5 Encodings: TeX, pMML, png See also: Annotations for §14.10 and Ch.14

${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$ also satisfies (14.10.1)–(14.10.5).

14.10.6		$$P_{\nu }^{\mu +2}\left(x\right)+2(\mu +1)x{\left(x^2-1\right)}^{-1/2}{P}_{\nu }^{\mu +1}\left(x\right)-(\nu -\mu )(\nu +\mu +1)P_{\nu }^{\mu }\left(x\right)=0,$$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.10, §14.14 Permalink: http://dlmf.nist.gov/14.10.E6 Encodings: TeX, pMML, png See also: Annotations for §14.10 and Ch.14

14.10.7		$${\left(x^2-1\right)}^{1/2}{P}_{\nu }^{\mu +1}\left(x\right)-(\nu -\mu +1)P_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1)x{P}_{\nu }^{\mu }\left(x\right)=0.$$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.10, §14.21(iii) Permalink: http://dlmf.nist.gov/14.10.E7 Encodings: TeX, pMML, png See also: Annotations for §14.10 and Ch.14

$Q_{\nu }^{\mu }\left(x\right)$ also satisfies (14.10.6) and (14.10.7). In addition, $P_{\nu }^{\mu }\left(x\right)$ and $Q_{\nu }^{\mu }\left(x\right)$ satisfy (14.10.3)–(14.10.5).