§14.10 Recurrence Relations and Derivatives

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

14.10.1 ${\mathop{\mathsf{P}^{{\mu+2}}_{{\nu}}\/}\nolimits\!\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{{-1/2}}\mathop{\mathsf{P}^{{\mu+1}}_{{\nu}}\/}\nolimits\!\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=0,$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.10.2 ${\left(1-x^{2}\right)^{{1/2}}\mathop{\mathsf{P}^{{\mu+1}}_{{\nu}}\/}\nolimits\!\left(x\right)-(\nu-\mu+1)\mathop{\mathsf{P}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)}+(\nu+\mu+1)x\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=0,$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.10.3 ${(\nu-\mu+2)\mathop{\mathsf{P}^{{\mu}}_{{\nu+2}}\/}\nolimits\!\left(x\right)-(2\nu+3)x\mathop{\mathsf{P}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)}+(\nu+\mu+1)\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=0,$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.10.4 $\left(1-x^{2}\right)\frac{d\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)}{dx}={(\mu-\nu-1)\mathop{\mathsf{P}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)+(\nu+1)x\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.10.5 $\left(1-x^{2}\right)\frac{d\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)}{dx}=(\nu+\mu)\mathop{\mathsf{P}^{{\mu}}_{{\nu-1}}\/}\nolimits\!\left(x\right)-\nu x\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

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

14.10.6 ${\mathop{P^{{\mu+2}}_{{\nu}}\/}\nolimits\!\left(x\right)+2(\mu+1)x\left(x^{2}-1\right)^{{-1/2}}\mathop{P^{{\mu+1}}_{{\nu}}\/}\nolimits\!\left(x\right)}-(\nu-\mu)(\nu+\mu+1)\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=0,$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

14.10.7 ${\left(x^{2}-1\right)^{{1/2}}\mathop{P^{{\mu+1}}_{{\nu}}\/}\nolimits\!\left(x\right)-(\nu-\mu+1)\mathop{P^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)}+(\nu+\mu+1)x\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=0.$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

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