§14.3 Definitions and Hypergeometric Representations

Keywords:: Ferrers functions, associated Legendre functions, hypergeometric function
Referenced by:: §14.19(ii), §14.5(v)
Permalink:: http://dlmf.nist.gov/14.3

§14.3(i) Interval $-1<x<1$
§14.3(ii) Interval $1<x<\infty$
§14.3(iii) Alternative Hypergeometric Representations
§14.3(iv) Relations to Other Functions

§14.3(i) Interval $-1<x<1$

Notes:: See Olver (1997b, pp. 159, 186). The version of (14.3.4) given in Hobson (1931, p. 386) has an error. For (14.3.5) use (14.3.1) and (14.9.3).
Keywords:: Ferrers functions, hypergeometric function
Referenced by:: ¶ ‣ §10.22(iv), §10.43(iv), §14.20(i), §14.23, §14.32, §15.9(iv), ¶ ‣ §18.11(i), ¶ ‣ §18.17(iii), §29.8, §30.4(i), §30.5, §30.8(i)
Permalink:: http://dlmf.nist.gov/14.3.i

The following are real-valued solutions of (14.2.2) when $\mu$ , $\nu\in\Real$ and $x\in(-1,1)$ .

¶ Ferrers Function of the First Kind

Keywords:: Ferrers functions

14.3.1 $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(\frac{1+x}{1-x}\right)^{{\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$

Defines:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind
Symbols:: $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.1.2 (modified)
Referenced by:: §14.11, §14.15(i), ¶ ‣ §14.3(i), §14.3(i)
Permalink:: http://dlmf.nist.gov/14.3.E1
Encodings:: TeX, pMML, png

¶ Ferrers Function of the Second Kind

Keywords:: Ferrers functions, Olver’s hypergeometric function

14.3.2 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\pi}{2\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}\left(\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)\left(\frac{1+x}{1-x}\right)^{{\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}\left(\frac{1-x}{1+x}\right)^{{\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;1+\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)\right).$

Defines:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: ¶ ‣ §14.3(i)
Permalink:: http://dlmf.nist.gov/14.3.E2
Encodings:: TeX, pMML, png

Here and elsewhere in this chapter

14.3.3 $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;x\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(c\right)}\mathop{F\/}\nolimits\!\left(a,b;c;x\right)$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/14.3.E3
Encodings:: TeX, pMML, png

is Olver’s hypergeometric function (§15.1).

$\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ exists for all values of $\mu$ and $\nu$ . $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ is undefined when $\mu+\nu=-1,-2,-3,\dots$ .

When $\mu=m=0,1,2,\ldots$ , (14.3.1) reduces to

14.3.4 $\mathop{\mathsf{P}^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)=(-1)^{m}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}{2^{m}\mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}\left(1-x^{2}\right)^{{m/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right);$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
Referenced by:: §14.3(i)
Permalink:: http://dlmf.nist.gov/14.3.E4
Encodings:: TeX, pMML, png

equivalently,

14.3.5 $\mathop{\mathsf{P}^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)=(-1)^{m}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}\left(\frac{1-x}{1+x}\right)^{{m/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
Referenced by:: §14.3(i)
Permalink:: http://dlmf.nist.gov/14.3.E5
Encodings:: TeX, pMML, png

When $\mu=m$ ( $\in\Integer$ ) (14.3.2) is replaced by its limiting value; see Hobson (1931, §132) for details. See also (14.3.12)–(14.3.14) for this case.

§14.3(ii) Interval $1<x<\infty$

Notes:: See Olver (1997b, p. 170). For (14.3.8) see Olver (1997b, p. 159, Eq. (9.05)).
Keywords:: associated Legendre functions, hypergeometric function
Referenced by:: ¶ ‣ §10.22(iv), ¶ ‣ §10.22(iv), ¶ ‣ §18.17(iii), ¶ ‣ §18.30, ¶ ‣ §30.6
Permalink:: http://dlmf.nist.gov/14.3.ii

¶ Associated Legendre Function of the First Kind

Keywords:: associated Legendre functions

14.3.6 $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(\frac{x+1}{x-1}\right)^{{\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$

Defines:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind
Symbols:: $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $z$ : complex variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.21(i), §14.21(iii), ¶ ‣ §14.3(ii), §14.3(iv), §15.9(iv)
Permalink:: http://dlmf.nist.gov/14.3.E6
Encodings:: TeX, pMML, png

¶ Associated Legendre Function of the Second Kind

Keywords:: Olver’s associated Legendre function, associated Legendre equation, associated Legendre functions

14.3.7 $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=e^{{\mu\pi i}}\frac{\pi^{{1/2}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\left(x^{2}-1\right)^{{\mu/2}}}{2^{{\nu+1}}x^{{\nu+\mu+1}}}\mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),$ $\mu+\nu\neq-1,-2,-3,\dots$ .

Defines:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $z$ : complex variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.1.3 (modified)
Permalink:: http://dlmf.nist.gov/14.3.E7
Encodings:: TeX, pMML, png

When $\mu=m=1,2,3,\dots$ , (14.3.6) reduces to

14.3.8 $\mathop{P^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}{2^{m}\mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}\left(x^{2}-1\right)^{{m/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree
Referenced by:: §14.3(ii)
Permalink:: http://dlmf.nist.gov/14.3.E8
Encodings:: TeX, pMML, png

As standard solutions of (14.2.2) we take the pair $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ , where

14.3.9 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(\frac{x-1}{x+1}\right)^{{\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.3.E9
Encodings:: TeX, pMML, png

and

14.3.10 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=e^{{-\mu\pi i}}\frac{\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}.$

Defines:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $e$ : base of exponential function, $x$ : real variable, $z$ : complex variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.20(i), §14.21(i), §14.21(iii), §14.6(i)
Permalink:: http://dlmf.nist.gov/14.3.E10
Encodings:: TeX, pMML, png

Like $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ , but unlike $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ is real-valued when $\nu$ , $\mu\in\Real$ and $x\in(1,\infty)$ , and is defined for all values of $\nu$ and $\mu$ . The notation $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ is due to Olver (1997b, pp. 170 and 178).

§14.3(iii) Alternative Hypergeometric Representations

Notes:: For (14.3.11) and (14.3.12) see Olver (1997b, p. 187). For (14.3.16)–(14.3.20) see Erdélyi et al. (1953a, pp. 123–139).
Keywords:: Ferrers functions, associated Legendre functions, hypergeometric function
Referenced by:: §14.32, §15.9(iv)
Permalink:: http://dlmf.nist.gov/14.3.iii

14.3.11 $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $w_{2}$ : function $w_{1}$ ,
A&S Ref:: 8.1.4 (modified)
Referenced by:: §10.59, §14.23, §14.3(iii), §14.5(i)
Permalink:: http://dlmf.nist.gov/14.3.E11
Encodings:: TeX, pMML, png

14.3.12 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=-\tfrac{1}{2}\pi\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\tfrac{1}{2}\pi\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x),$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $w_{2}$ : function $w_{1}$ ,
A&S Ref:: 8.1.7 (modified)
Referenced by:: §14.23, ¶ ‣ §14.3(i), §14.3(iii)
Permalink:: http://dlmf.nist.gov/14.3.E12
Encodings:: TeX, pMML, png

where

14.3.13 $w_{1}(\nu,\mu,x)=\frac{2^{{\mu}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)}\left(1-x^{2}\right)^{{-\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+\tfrac{1}{2};\tfrac{1}{2};x^{2}\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $w_{2}$ : function $w_{1}$ ,
Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/14.3.E13
Encodings:: TeX, pMML, png

14.3.14 $w_{2}(\nu,\mu,x)=\frac{2^{{\mu}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}x\left(1-x^{2}\right)^{{-\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2};x^{2}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree and $w_{2}$ : function $w_{1}$ ,
Referenced by:: §10.59, ¶ ‣ §14.3(i), §14.5(i)
Permalink:: http://dlmf.nist.gov/14.3.E14
Encodings:: TeX, pMML, png

14.3.15 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=2^{{-\mu}}\left(x^{2}-1\right)^{{\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.21(iii)
Permalink:: http://dlmf.nist.gov/14.3.E15
Encodings:: TeX, pMML, png

14.3.16 $\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{2^{{\nu}}\pi^{{1/2}}x^{{\nu-\mu}}\left(x^{2}-1\right)^{{\mu/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac{\pi^{{1/2}}\left(x^{2}-1\right)^{{\mu/2}}}{2^{{\nu+1}}\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)x^{{\nu+\mu+1}}}\mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.1.5 (modified)
Referenced by:: §14.3(iii)
Permalink:: http://dlmf.nist.gov/14.3.E16
Encodings:: TeX, pMML, png

14.3.17 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\pi\left(x^{2}-1\right)^{{\mu/2}}}{2^{{\mu}}}\left(\frac{\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}-\frac{x\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\mu-\frac{1}{2}\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.3.E17
Encodings:: TeX, pMML, png

14.3.18 $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=2^{{-\mu}}x^{{\nu-\mu}}\left(x^{2}-1\right)^{{\mu/2}}\mathop{\mathbf{F}\/}\nolimits\!\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\mu+1;1-\frac{1}{x^{2}}\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.3.E18
Encodings:: TeX, pMML, png

14.3.19 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{2^{{\nu}}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)(x+1)^{{\mu/2}}}{(x-1)^{{(\mu/2)+\nu+1}}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,\nu+\mu+1;2\nu+2;\frac{2}{1-x}\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.3.E19
Encodings:: TeX, pMML, png

14.3.20 $\frac{2\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{\pi}\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{(x+1)^{{\mu/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)(x-1)^{{\mu/2}}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{(x-1)^{{\mu/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)(x+1)^{{\mu/2}}}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.1.6 (modified)
Referenced by:: §14.21(iii), §14.3(iii)
Permalink:: http://dlmf.nist.gov/14.3.E20
Encodings:: TeX, pMML, png

For further hypergeometric representations of $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ see Erdélyi et al. (1953a, pp. 123–139), Andrews et al. (1999, §3.1), Magnus et al. (1966, pp. 153–163), and §15.8(iii).

§14.3(iv) Relations to Other Functions

Notes:: For (14.3.21) combine (14.3.22) and (14.23.4). For (14.3.22) see Erdélyi et al. (1953a, p. 175). For (14.3.23) see (14.3.6) and Olver (1997b, p. 167).
Keywords:: Gegenbauer function, Jacobi function, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.3.iv

In terms of the Gegenbauer function $\mathop{C^{{(\beta)}}_{{\alpha}}\/}\nolimits\!\left(x\right)$ and the Jacobi function $\mathop{\phi^{{(\alpha,\beta)}}_{{\lambda}}\/}\nolimits\!\left(t\right)$ (§§15.9(iii), 15.9(ii)):

14.3.21 $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{2^{{\mu}}\mathop{\Gamma\/}\nolimits\!\left(1-2\mu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)\left(1-x^{2}\right)^{{\mu/2}}}\mathop{C^{{(\frac{1}{2}-\mu)}}_{{\nu+\mu}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.3(iv), §18.10(i), ¶ ‣ §18.11(i)
Permalink:: http://dlmf.nist.gov/14.3.E21
Encodings:: TeX, pMML, png

14.3.22 $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{2^{{\mu}}\mathop{\Gamma\/}\nolimits\!\left(1-2\mu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)\left(x^{2}-1\right)^{{\mu/2}}}\mathop{C^{{(\frac{1}{2}-\mu)}}_{{\nu+\mu}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.3(iv), ¶ ‣ §18.11(i)
Permalink:: http://dlmf.nist.gov/14.3.E22
Encodings:: TeX, pMML, png

14.3.23 $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{{\mu/2}}\mathop{\phi^{{(-\mu,\mu)}}_{{-i(2\nu+1)}}\/}\nolimits\!\left(\mathop{\mathrm{arcsinh}\/}\nolimits\!\left((\tfrac{1}{2}x-\tfrac{1}{2})^{{\ifrac{1}{2}}}\right)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\phi^{{(\alpha,\beta)}}_{{\lambda}}\/}\nolimits\!\left(t\right)$ : Jacobi function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.3(iv)
Permalink:: http://dlmf.nist.gov/14.3.E23
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Compare also (18.11.1).