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DLMF: §14.6 Integer Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions

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14 Legendre and Related Functions Real Arguments 14.5 Special Values 14.7 Integer Degree and Order

§14.6 Integer Order

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Keywords:: Ferrers functions, associated Legendre functions, integer order
Permalink:: http://dlmf.nist.gov/14.6
See also:: Annotations for Ch.14

Contents

§14.6(i) Nonnegative Integer Orders

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Notes:: See Erdélyi et al. (1953a, pp. 148–149). For (14.6.5) combine (14.3.10) and (14.6.4).
Permalink:: http://dlmf.nist.gov/14.6.i
See also:: Annotations for §14.6 and Ch.14

For $m=0,1,2,\dots$ ,

14.6.1		$\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)$	$\displaystyle=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathsf{% P}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}},$
ⓘ 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 $x$ , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : 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		$\displaystyle\mathsf{Q}^{m}_{\nu}\left(x\right)$	$\displaystyle=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}\mathsf{% Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}.$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : 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		$\displaystyle P^{m}_{\nu}\left(x\right)$	$\displaystyle=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}P_{\nu}\left(x% \right)}{{\mathrm{d}x}^{m}},$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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		$\displaystyle Q^{m}_{\nu}\left(x\right)$	$\displaystyle=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}Q_{\nu}\left(x% \right)}{{\mathrm{d}x}^{m}},$
ⓘ Symbols: $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : 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		${\left(\nu+1\right)_{m}}\boldsymbol{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(x% ^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}\boldsymbol{Q}_{\nu}\left(x\right)}{{% \mathrm{d}x}^{m}}.$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : 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

§14.6(ii) Negative Integer Orders

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Keywords:: Ferrers functions, associated Legendre functions, integer order
Notes:: See Erdélyi et al. (1953a, p. 149).
Referenced by:: Erratum (V1.1.1) for Subsection 14.6(ii)
Permalink:: http://dlmf.nist.gov/14.6.ii
Addition (effective with 1.1.1):: A sentence was added at the end of the subsection indicating that for generalizations, see Cohl and Costas-Santos (2020).
See also:: Annotations for §14.6 and Ch.14

For $m=1,2,3,\dots$ ,

14.6.6		$\displaystyle\mathsf{P}^{-m}_{\nu}\left(x\right)$	$\displaystyle=\left(1-x^{2}\right)^{-m/2}\int_{x}^{1}\!\dots\!\int_{x}^{1}% \mathsf{P}_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^{m}.$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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		$\displaystyle P^{-m}_{\nu}\left(x\right)$	$\displaystyle=\left(x^{2}-1\right)^{-m/2}\int_{1}^{x}\!\dots\!\int_{1}^{x}P_{% \nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^{m},$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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		$\displaystyle Q^{-m}_{\nu}\left(x\right)$	$\displaystyle=(-1)^{m}\left(x^{2}-1\right)^{-m/2}\*\int_{x}^{\infty}\!\dots\!% \int_{x}^{\infty}Q_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^{m}.$
ⓘ Symbols: $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : 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). For generalizations see Cohl and Costas-Santos (2020).

14.5 Special Values 14.7 Integer Degree and Order

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