14 Legendre and Related Functions Real Arguments 14.2 Differential Equations 14.4 Graphics

§14.3 Definitions and Hypergeometric Representations

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Keywords:: Ferrers functions, associated Legendre functions, definitions, hypergeometric function, hypergeometric representations, relations to other functions
Referenced by:: §14.19(ii), §14.5(v)
Permalink:: http://dlmf.nist.gov/14.3
See also:: Annotations for Ch.14

§14.3(i) Interval $-1<x<1$

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Keywords:: Ferrers functions, hypergeometric function, relations to other functions
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).
Referenced by:: §10.22(iv), §10.43(iv), §14.18(iii), §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
See also:: Annotations for §14.3 and Ch.14

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

Ferrers Function of the First Kind

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Keywords:: Ferrers functions, of the first kind
See also:: Annotations for §14.3(i), §14.3 and Ch.14

14.3.1		$\mathsf{P}^{\mu}_{\nu}\left(x\right)=\left(\frac{1+x}{1-x}\right)^{\mu/2}% \mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$
ⓘ Defines: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind Symbols: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $1<x<1$ A&S Ref: 8.1.2 (modified) Referenced by: §14.11, §14.15(i), §14.3(i), §14.3(i), §15.9(iv), §15.9(iv) Permalink: http://dlmf.nist.gov/14.3.E1 Encodings: TeX, pMML, png See also: Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

Ferrers Function of the Second Kind

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Keywords:: Ferrers functions, Olver’s hypergeometric function, of the second kind
See also:: Annotations for §14.3(i), §14.3 and Ch.14

14.3.2		$\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi}{2\sin\left(\mu\pi\right)}\left% (\cos\left(\mu\pi\right)\left(\frac{1+x}{1-x}\right)^{\mu/2}\mathbf{F}\left(% \nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)}\left(\frac{1-x}{1+x}\right)^{\mu/2}% \mathbf{F}\left(\nu+1,-\nu;1+\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)\right).$
ⓘ Defines: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $1<x<1$ Referenced by: §14.3(i) Permalink: http://dlmf.nist.gov/14.3.E2 Encodings: TeX, pMML, png See also: Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

Here and elsewhere in this chapter

14.3.3		$\mathbf{F}\left(a,b;c;x\right)=\frac{1}{\Gamma\left(c\right)}F\left(a,b;c;x\right)$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function and $x$ : real variable $1<x<1$ Permalink: http://dlmf.nist.gov/14.3.E3 Encodings: TeX, pMML, png See also: Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

is Olver’s hypergeometric function (§15.1).

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

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

14.3.4		$\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{2% ^{m}\Gamma\left(\nu-m+1\right)}\left(1-x^{2}\right)^{m/2}\mathbf{F}\left(\nu+m% +1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right);$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $m$ : nonnegative integer, $\nu$ : general degree and $x$ : real variable $1<x<1$ Referenced by: §14.3(i) Permalink: http://dlmf.nist.gov/14.3.E4 Encodings: TeX, pMML, png See also: Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

equivalently,

14.3.5		$\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{% \Gamma\left(\nu-m+1\right)}\left(\frac{1-x}{1+x}\right)^{m/2}\mathbf{F}\left(% \nu+1,-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $m$ : nonnegative integer, $\nu$ : general degree and $x$ : real variable $1<x<1$ Referenced by: §14.3(i) Permalink: http://dlmf.nist.gov/14.3.E5 Encodings: TeX, pMML, png See also: Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

When $\mu=m$ ( $\in\mathbb{Z}$ ) (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 $1<x<\infty$

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Keywords:: associated Legendre functions, definitions, hypergeometric function, hypergeometric representations, relations to other functions
Notes:: See Olver (1997b, p. 170). For (14.3.8) see Olver (1997b, p. 159, Eq. (9.05)).
Referenced by:: §10.22(iv), §10.22(iv), §14.18(iii), §18.17(iii), §30.6
Permalink:: http://dlmf.nist.gov/14.3.ii
See also:: Annotations for §14.3 and Ch.14

The following are solutions of (14.2.2) when $\mu$ , $\nu\in\mathbb{R}$ and $x>1$ .

Associated Legendre Function of the First Kind

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Keywords:: associated Legendre functions, of the first kind
See also:: Annotations for §14.3(ii), §14.3 and Ch.14

14.3.6		$P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ 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 See also: Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

Associated Legendre Function of the Second Kind

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Keywords:: Olver’s, Olver’s associated Legendre function, associated Legendre equation, associated Legendre functions, of the second kind, standard solutions
See also:: Annotations for §14.3(ii), §14.3 and Ch.14

14.3.7		$Q^{\mu}_{\nu}\left(x\right)=e^{\mu\pi i}\frac{\pi^{1/2}\Gamma\left(\nu+\mu+1% \right)\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}x^{\nu+\mu+1}}\mathbf{F}\left(% \tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}% ;\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),$
$\mu+\nu\neq-1,-2,-3,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ A&S Ref: 8.1.3 (modified) Permalink: http://dlmf.nist.gov/14.3.E7 Encodings: TeX, pMML, png See also: Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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

14.3.8		$P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(% \nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $m$ : nonnegative integer, $\nu$ : general degree and $x$ : real variable $x>1$ Referenced by: §14.3(ii) Permalink: http://dlmf.nist.gov/14.3.E8 Encodings: TeX, pMML, png See also: Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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

14.3.9		$P^{-\mu}_{\nu}\left(x\right)=\left(\frac{x-1}{x+1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ Permalink: http://dlmf.nist.gov/14.3.E9 Encodings: TeX, pMML, png See also: Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

and

14.3.10		$\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i}\frac{Q^{\mu}_{\nu}\left% (x\right)}{\Gamma\left(\nu+\mu+1\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ 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 See also: Annotations for §14.3(ii), §14.3(ii), §14.3 and Ch.14

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

§14.3(iii) Alternative Hypergeometric Representations

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Keywords:: Ferrers functions, associated Legendre functions, definitions, hypergeometric function, hypergeometric representations, relations to other functions
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).
Referenced by:: §14.32, §15.9(iv), Erratum (V1.1.3) for References
Permalink:: http://dlmf.nist.gov/14.3.iii
Addition (effective with 1.1.3):: A sentence was added at the end of this subsection indicating that for further hypergeometric representations of $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ see Cohl et al. (2021).
See also:: Annotations for §14.3 and Ch.14

14.3.11	$\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)$	$\displaystyle=\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\sin% \left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree, $x$ : real variable $1<x<1$ 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 See also: Annotations for §14.3(iii), §14.3 and Ch.14
14.3.12	$\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)$	$\displaystyle=-\tfrac{1}{2}\pi\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(% \nu,\mu,x)+\tfrac{1}{2}\pi\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,% \mu,x),$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree, $x$ : real variable $1<x<1$ 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 See also: Annotations for §14.3(iii), §14.3 and Ch.14

where

14.3.13	$\displaystyle w_{1}(\nu,\mu,x)$	$\displaystyle=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{% 2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)}\left(1-x^{2}% \right)^{-\mu/2}\mathbf{F}\left(-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}% \nu-\tfrac{1}{2}\mu+\tfrac{1}{2};\tfrac{1}{2};x^{2}\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 See also: Annotations for §14.3(iii), §14.3 and Ch.14
14.3.14	$\displaystyle w_{2}(\nu,\mu,x)$	$\displaystyle=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}% {\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}x\left(1-x^{2}% \right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,% \tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2};x^{2}\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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 See also: Annotations for §14.3(iii), §14.3 and Ch.14

14.3.15		$P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}% \left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.3.E15 Encodings: TeX, pMML, png See also: Annotations for §14.3(iii), §14.3 and Ch.14

14.3.16		$\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\pi^{1/2}x^{% \nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)}\mathbf{F}% \left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1% }{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac{\pi^{1/2}\left(x^{2}-1% \right)^{\mu/2}}{2^{\nu+1}\Gamma\left(\mu-\nu\right)x^{\nu+\mu+1}}\mathbf{F}% \left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac% {1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ A&S Ref: 8.1.5 (modified) Referenced by: §14.3(iii) Permalink: http://dlmf.nist.gov/14.3.E16 Encodings: TeX, pMML, png See also: Annotations for §14.3(iii), §14.3 and Ch.14

14.3.17		$P^{-\mu}_{\nu}\left(x\right)=\frac{\pi\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}}% \left(\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac% {1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\mu-% \frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1% \right)}-\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2},% \frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{\Gamma\left(\frac{1}% {2}\mu-\frac{1}{2}\nu\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}% {2}\right)}\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ Permalink: http://dlmf.nist.gov/14.3.E17 Encodings: TeX, pMML, png See also: Annotations for §14.3(iii), §14.3 and Ch.14

14.3.18	$\displaystyle P^{-\mu}_{\nu}\left(x\right)$	$\displaystyle=2^{-\mu}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}\left(% \tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};% \mu+1;1-\frac{1}{x^{2}}\right),$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ Permalink: http://dlmf.nist.gov/14.3.E18 Encodings: TeX, pMML, png See also: Annotations for §14.3(iii), §14.3 and Ch.14
14.3.19	$\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$	$\displaystyle=\frac{2^{\nu}\Gamma\left(\nu+1\right)(x+1)^{\mu/2}}{(x-1)^{(\mu/% 2)+\nu+1}}\mathbf{F}\left(\nu+1,\nu+\mu+1;2\nu+2;\frac{2}{1-x}\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ Permalink: http://dlmf.nist.gov/14.3.E19 Encodings: TeX, pMML, png See also: Annotations for §14.3(iii), §14.3 and Ch.14

14.3.20		$\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=% \frac{(x+1)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)(x-1)^{\mu/2}}\mathbf{F}\left% (\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{(x-1)^{\mu/2}}{% \Gamma\left(\nu-\mu+1\right)(x+1)^{\mu/2}}\mathbf{F}\left(\nu+1,-\nu;\mu+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ 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 See also: Annotations for §14.3(iii), §14.3 and Ch.14

For further hypergeometric representations of $P^{\mu}_{\nu}\left(x\right)$ and $Q^{\mu}_{\nu}\left(x\right)$ 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). For further hypergeometric representations of $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ see Cohl et al. (2021).

§14.3(iv) Relations to Other Functions

ⓘ

Keywords:: Gegenbauer function, Jacobi function, associated Legendre functions, relation to associated Legendre functions, 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).
Referenced by:: Erratum (V1.0.28) for Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function, Erratum (V1.1.1) for Subsection 14.3(iv)
Permalink:: http://dlmf.nist.gov/14.3.iv
Addition (effective with 1.1.1):: A sentence was added at the end of this subsection indicating that from (15.9.15), it follows that $1-2\mu=0,-1,-2,\dots$ and $\nu+\mu+1=0,-1,-2,\dots$ are removable singularities.
See also:: Annotations for §14.3 and Ch.14

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

14.3.21	$\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)$	$\displaystyle=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(1-x^{2}% \right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $C^{(\NVar{\lambda})}_{\NVar{\alpha}}\left(\NVar{z}\right)$ : Gegenbauer function, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $1<x<1$ Referenced by: §14.3(iv), §14.3(iv), §18.10(i), §18.11(i) Permalink: http://dlmf.nist.gov/14.3.E21 Encodings: TeX, pMML, png See also: Annotations for §14.3(iv), §14.3 and Ch.14
14.3.22	$\displaystyle P^{\mu}_{\nu}\left(x\right)$	$\displaystyle=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(x^{2}-1% \right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $C^{(\NVar{\lambda})}_{\NVar{\alpha}}\left(\NVar{z}\right)$ : Gegenbauer function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ Referenced by: §14.3(iv), §14.3(iv), §18.11(i) Permalink: http://dlmf.nist.gov/14.3.E22 Encodings: TeX, pMML, png See also: Annotations for §14.3(iv), §14.3 and Ch.14
14.3.23	$\displaystyle P^{\mu}_{\nu}\left(x\right)$	$\displaystyle=\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{% \mu/2}\phi^{(-\mu,\mu)}_{-\mathrm{i}(2\nu+1)}\left(\operatorname{arcsinh}\left% ((\tfrac{1}{2}x-\tfrac{1}{2})^{\ifrac{1}{2}}\right)\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda}}\left(\NVar{t}\right)$ : Jacobi function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\mu$ : general order, $\nu$ : general degree and $x$ : real variable $x>1$ Referenced by: §14.3(iv) Permalink: http://dlmf.nist.gov/14.3.E23 Encodings: TeX, pMML, png See also: Annotations for §14.3(iv), §14.3 and Ch.14

Compare also (18.11.1). From (15.9.15) it follows that $1-2\mu=0,-1,-2,\dots$ and $\nu+\mu+1=0,-1,-2,\dots$ are removable singularities of the right-hand sides of (14.3.21) and (14.3.22).

14.2 Differential Equations 14.4 Graphics

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§14.3 Definitions and Hypergeometric Representations

Contents

§14.3(i) Interval $-1<x<1$

Ferrers Function of the First Kind

Ferrers Function of the Second Kind

§14.3(ii) Interval $1<x<\infty$

Associated Legendre Function of the First Kind

Associated Legendre Function of the Second Kind

§14.3(iii) Alternative Hypergeometric Representations

§14.3(iv) Relations to Other Functions

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§14.3 Definitions and Hypergeometric Representations

Contents

§14.3(i) Interval −1<x<1

Ferrers Function of the First Kind

Ferrers Function of the Second Kind

§14.3(ii) Interval 1<x<∞

Associated Legendre Function of the First Kind

Associated Legendre Function of the Second Kind

§14.3(iii) Alternative Hypergeometric Representations

§14.3(iv) Relations to Other Functions

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§14.3(i) Interval $-1<x<1$

§14.3(ii) Interval $1<x<\infty$