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

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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 ${\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,$

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

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

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

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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 with respect to , : 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 $\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}=(\nu+\mu)\mathsf{P}^{\mu}_{\nu-1}\left(x\right)-\nu x\mathsf{P}^{% \mu}_{\nu}\left(x\right).$

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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 with respect to , : 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 ${P^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(x^{2}-1\right)^{-1/2}P^{\mu+1}_{% \nu}\left(x\right)}-(\nu-\mu)(\nu+\mu+1)P^{\mu}_{\nu}\left(x\right)=0,$

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : 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 ${\left(x^{2}-1\right)^{1/2}P^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)P^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)xP^{\mu}_{\nu}\left(x\right)=0.$

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : 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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