§14.6 Integer Order

For $m=0,1,2,\mathrm{\dots }$,

14.6.1		${\mathsf{P}}_{\nu }^m\left(x\right)$	$={(-1)}^m{\left(1-x^2\right)}^{m/2}{\displaystyle \frac{d^m{\mathsf{P}}_{\nu }\left(x\right)}{{dx}^m}},$
ⓘ Symbols: : Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree A&S Ref: 8.6.6 Permalink: http://dlmf.nist.gov/14.6.E1 Encodings: TeX, pMML, png See also: Annotations for §14.6(i), §14.6 and Ch.14
14.6.2		${\mathsf{Q}}_{\nu }^m\left(x\right)$	$={(-1)}^m{\left(1-x^2\right)}^{m/2}{\displaystyle \frac{d^m{\mathsf{Q}}_{\nu }\left(x\right)}{{dx}^m}}.$
ⓘ Symbols: : Ferrers function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree A&S Ref: 8.6.7 Permalink: http://dlmf.nist.gov/14.6.E2 Encodings: TeX, pMML, png See also: Annotations for §14.6(i), §14.6 and Ch.14

14.6.3		$P_{\nu }^m\left(x\right)$	$={\left(x^2-1\right)}^{m/2}{\displaystyle \frac{d^m{P}_{\nu }\left(x\right)}{{dx}^m}},$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree A&S Ref: 8.6.6 Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.6.E3 Encodings: TeX, pMML, png See also: Annotations for §14.6(i), §14.6 and Ch.14
14.6.4		$Q_{\nu }^m\left(x\right)$	$={\left(x^2-1\right)}^{m/2}{\displaystyle \frac{d^m{Q}_{\nu }\left(x\right)}{{dx}^m}},$
ⓘ Symbols: $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree A&S Ref: 8.6.7 Referenced by: §14.6(i) Permalink: http://dlmf.nist.gov/14.6.E4 Encodings: TeX, pMML, png See also: Annotations for §14.6(i), §14.6 and Ch.14

14.6.5		$${\left(\nu +1\right)}_m{\mathit{Q}}_{\nu }^m\left(x\right)={(-1)}^m{\left(x^2-1\right)}^{m/2}\frac{d^m{\mathit{Q}}_{\nu }\left(x\right)}{{dx}^m}.$$
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree Referenced by: §14.21(iii), §14.6(i) Permalink: http://dlmf.nist.gov/14.6.E5 Encodings: TeX, pMML, png See also: Annotations for §14.6(i), §14.6 and Ch.14

For $m=1,2,3,\mathrm{\dots }$,

14.6.6		${\mathsf{P}}_{\nu }^{-m}\left(x\right)$	$={\left(1-x^2\right)}^{-m/2}{\displaystyle {\int }_x^1}\mathrm{\dots }{\displaystyle {\int }_x^1}{\mathsf{P}}_{\nu }\left(x\right){\left(dx\right)}^m.$
ⓘ Symbols: : Ferrers function of the first kind, $dx$: differential of $x$, $\int$: integral, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree Referenced by: §14.12(i), Erratum (V1.0.28) for Equation (14.6.6) Permalink: http://dlmf.nist.gov/14.6.E6 Encodings: TeX, pMML, png Clarification (effective with 1.0.28): The right-hand side has been corrected by replacing the Legendre function $P_{\nu }\left(x\right)$ with the Ferrers function ${\mathsf{P}}_{\nu }\left(x\right)$. See also: Annotations for §14.6(ii), §14.6 and Ch.14
14.6.7		$P_{\nu }^{-m}\left(x\right)$	$={\left(x^2-1\right)}^{-m/2}{\displaystyle {\int }_1^x}\mathrm{\dots }{\displaystyle {\int }_1^x}{P}_{\nu }\left(x\right){\left(dx\right)}^m,$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$, $\int$: integral, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree A&S Ref: 8.14.17 Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.6.E7 Encodings: TeX, pMML, png See also: Annotations for §14.6(ii), §14.6 and Ch.14
14.6.8		$Q_{\nu }^{-m}\left(x\right)$	$={(-1)}^m{\left(x^2-1\right)}^{-m/2}{\displaystyle {\int }_x^{\mathrm{\infty }}}\mathrm{\dots }{\displaystyle {\int }_x^{\mathrm{\infty }}}{Q}_{\nu }\left(x\right){\left(dx\right)}^m.$
ⓘ Symbols: $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $dx$: differential of $x$, $\int$: integral, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree A&S Ref: 8.14.18 Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.6.E8 Encodings: TeX, pMML, png See also: Annotations for §14.6(ii), §14.6 and Ch.14

For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13).