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

§14.6 Integer Order

Keywords:: Ferrers functions, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.6

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

§14.6(i) Nonnegative Integer Orders

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

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

14.6.1 $\mathop{\mathsf{P}^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)=(-1)^{m}\left(1-x% ^{2}\right)^{{m/2}}\frac{{d}^{m}\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(% x\right)}{{dx}^{m}},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\frac{df}{dx}$ : derivative of with respect to , : 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

14.6.2 $\mathop{\mathsf{Q}^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)=(-1)^{m}\left(1-x% ^{2}\right)^{{m/2}}\frac{{d}^{m}\mathop{\mathsf{Q}_{{\nu}}\/}\nolimits\!\left(% x\right)}{{dx}^{m}}.$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\frac{df}{dx}$ : derivative of with respect to , : 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

14.6.3 $\mathop{P^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(x^{2}-1\right)^{{m/2% }}\frac{{d}^{m}\mathop{P_{{\nu}}\/}\nolimits\!\left(x\right)}{{dx}^{m}},$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\frac{df}{dx}$ : derivative of with respect to , : 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

14.6.4 $\mathop{Q^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(x^{2}-1\right)^{{m/2% }}\frac{{d}^{m}\mathop{Q_{{\nu}}\/}\nolimits\!\left(x\right)}{{dx}^{m}},$

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\frac{df}{dx}$ : derivative of with respect to , : 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

14.6.5 $\left(\nu+1\right)_{{m}}\mathop{\boldsymbol{Q}^{{m}}_{{\nu}}\/}\nolimits\!% \left(x\right)=(-1)^{m}\left(x^{2}-1\right)^{{m/2}}\frac{{d}^{m}\mathop{% \boldsymbol{Q}_{{\nu}}\/}\nolimits\!\left(x\right)}{{dx}^{m}}.$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\frac{df}{dx}$ : derivative of with respect to , : 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

§14.6(ii) Negative Integer Orders

Notes:: See Erdélyi et al. (1953a, p. 149).
Keywords:: Ferrers functions, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.6.ii

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

14.6.6 $\mathop{\mathsf{P}^{{-m}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(1-x^{2}% \right)^{{-m/2}}\int_{x}^{1}\!\dots\!\int_{x}^{1}\mathop{P_{{\nu}}\/}\nolimits% \!\left(x\right)\left(\!dx\right)^{m}.$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : differential of , $\int$ : integral, : real variable, : nonnegative integer and $\nu$ : general degree
Referenced by:: ¶ ‣ §14.12(i)
Permalink:: http://dlmf.nist.gov/14.6.E6
Encodings:: TeX, pMML, png

14.6.7 $\mathop{P^{{-m}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(x^{2}-1\right)^{{-m% /2}}\int_{1}^{x}\!\dots\!\int_{1}^{x}\mathop{P_{{\nu}}\/}\nolimits\!\left(x% \right)\left(\!dx\right)^{m},$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : differential of , $\int$ : integral, : 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

14.6.8 $\mathop{Q^{{-m}}_{{\nu}}\/}\nolimits\!\left(x\right)=(-1)^{m}\left(x^{2}-1% \right)^{{-m/2}}\*\int_{x}^{{\infty}}\!\dots\!\int_{x}^{{\infty}}\mathop{Q_{{% \nu}}\/}\nolimits\!\left(x\right)\left(\!dx\right)^{m}.$

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, : differential of , $\int$ : integral, : 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

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

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§14.6 Integer Order

Contents

§14.6(i) Nonnegative Integer Orders

§14.6(ii) Negative Integer Orders