# §14.10 Recurrence Relations and Derivatives

 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: $\mathop{\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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
 14.10.2 ${\left(1-x^{2}\right)^{1/2}\mathop{\mathsf{P}^{\mu+1}_{\nu}\/}\nolimits\!\left% (x\right)-(\nu-\mu+1)\mathop{\mathsf{P}^{\mu}_{\nu+1}\/}\nolimits\!\left(x% \right)}+(\nu+\mu+1)x\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right% )=0,$
 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: $\mathop{\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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
 14.10.4 $\left(1-x^{2}\right)\frac{\mathrm{d}\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits% \!\left(x\right)}{\mathrm{d}x}={(\mu-\nu-1)\mathop{\mathsf{P}^{\mu}_{\nu+1}\/}% \nolimits\!\left(x\right)+(\nu+1)x\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!% \left(x\right)},$
 14.10.5 $\left(1-x^{2}\right)\frac{\mathrm{d}\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits% \!\left(x\right)}{\mathrm{d}x}=(\nu+\mu)\mathop{\mathsf{P}^{\mu}_{\nu-1}\/}% \nolimits\!\left(x\right)-\nu x\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!% \left(x\right).$

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

$\mathop{Q^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ 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).