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

§14.6(i) Nonnegative Integer Orders
§14.6(ii) Negative Integer Orders

§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 $\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}\mathsf{P}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}},$

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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 , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, : real variable, : 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 $\mathsf{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}\mathsf{Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}.$

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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 with respect to , $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, : real variable, : 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 $P^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}P_{% \nu}\left(x\right)}{{\mathrm{d}x}^{m}},$

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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 with respect to , $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, : real variable, : 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 $Q^{m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}Q_{% \nu}\left(x\right)}{{\mathrm{d}x}^{m}},$

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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 with respect to , $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, : real variable, : 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 ${\left(\nu+1\right)_{m}}\boldsymbol{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(x% ^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}\boldsymbol{Q}_{\nu}\left(x\right)}{{% \mathrm{d}x}^{m}}.$

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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 with respect to , $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : Olver’s associated Legendre function, : real variable, : 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 $\mathsf{P}^{-m}_{\nu}\left(x\right)=\left(1-x^{2}\right)^{-m/2}\int_{x}^{1}\!% \dots\!\int_{x}^{1}\mathsf{P}_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^% {m}.$

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : 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, : real variable, : 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 $P^{-m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{-m/2}\int_{1}^{x}\!\dots\!% \int_{1}^{x}P_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^{m},$

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, : real variable, : 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 $Q^{-m}_{\nu}\left(x\right)=(-1)^{m}\left(x^{2}-1\right)^{-m/2}\*\int_{x}^{% \infty}\!\dots\!\int_{x}^{\infty}Q_{\nu}\left(x\right)\left(\!\,\mathrm{d}x% \right)^{m}.$

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Symbols:: $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, : real variable, : 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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§14.6 Integer Order

Contents

§14.6(i) Nonnegative Integer Orders

§14.6(ii) Negative Integer Orders