14 Legendre and Related Functions Real Arguments 14.9 Connection Formulas 14.11 Derivatives with Respect to Degree or Order

§14.10 Recurrence Relations and Derivatives

ⓘ

Keywords:: Ferrers functions, associated Legendre functions, derivatives, recurrence relations
Notes:: See Erdélyi et al. (1953a, pp. 160–161).
Referenced by:: §14.17(i), §14.32
Permalink:: http://dlmf.nist.gov/14.10
See also:: Annotations for Ch.14

14.10.1		${\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		${\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)% \mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}% \left(x\right)=0,$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		${(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0,$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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		$\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x% \mathsf{P}^{\mu}_{\nu}\left(x\right)},$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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		$\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}=(\nu+\mu)\mathsf{P}^{\mu}_{\nu-1}\left(x\right)-\nu x\mathsf{P}^{% \mu}_{\nu}\left(x\right).$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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

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

14.10.6		${P^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(x^{2}-1\right)^{-1/2}P^{\mu+1}_{% \nu}\left(x\right)}-(\nu-\mu)(\nu+\mu+1)P^{\mu}_{\nu}\left(x\right)=0,$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		${\left(x^{2}-1\right)^{1/2}P^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)P^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)xP^{\mu}_{\nu}\left(x\right)=0.$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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

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

14.9 Connection Formulas 14.11 Derivatives with Respect to Degree or Order

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