§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 $f$ with respect to $x$ , $x$ : 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

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 $f$ with respect to $x$ , $x$ : 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

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 $f$ with respect to $x$ , $x$ : 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

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 $f$ with respect to $x$ , $x$ : 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

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 $f$ with respect to $x$ , $x$ : 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

§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, $dx$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $m$ : 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, $dx$ : differential of $x$ , $\int$ : integral, $x$ : 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

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, $dx$ : differential of $x$ , $\int$ : integral, $x$ : 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

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

§14.6 Integer Order

Contents

§14.6(i) Nonnegative Integer Orders

§14.6(ii) Negative Integer Orders