14 Legendre and Related Functions Real Arguments 14.8 Behavior at Singularities 14.10 Recurrence Relations and Derivatives

§14.9 Connection Formulas

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Referenced by:: §14.2(iv)
Permalink:: http://dlmf.nist.gov/14.9
See also:: Annotations for Ch.14

§14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$
§14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$
§14.9(iii) Connections Between $P^{\pm\mu}_{\nu}\left(x\right)$ , $P^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\boldsymbol{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)$
§14.9(iv) Whipple’s Formula

§14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$

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Keywords:: Ferrers functions, connection formulas
Notes:: See Olver (1997b, p. 186). For (14.9.3) and (14.9.4) use (14.9.1) and (14.9.2).
Permalink:: http://dlmf.nist.gov/14.9.i
See also:: Annotations for §14.9 and Ch.14

14.9.1		$\frac{\pi\sin\left(\mu\pi\right)}{2\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-% \mu}_{\nu}\left(x\right)=-\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)+\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.9(i) Permalink: http://dlmf.nist.gov/14.9.E1 Encodings: TeX, pMML, png See also: Annotations for §14.9(i), §14.9 and Ch.14

14.9.2		$\frac{2\sin\left(\mu\pi\right)}{\pi\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-% \mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{\mu% }_{\nu}\left(x\right)-\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.8(i), §14.9(i) Permalink: http://dlmf.nist.gov/14.9.E2 Encodings: TeX, pMML, png See also: Annotations for §14.9(i), §14.9 and Ch.14

14.9.3		$\mathsf{P}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{P}^{m}_{\nu}\left(x\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree Referenced by: §14.17(iii), §14.18(ii), §14.3(i), §14.6(ii), §14.9(i) Permalink: http://dlmf.nist.gov/14.9.E3 Encodings: TeX, pMML, png See also: Annotations for §14.9(i), §14.9 and Ch.14

14.9.4		$\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}\left(x\right),$
$\nu\neq m-1,m-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree Referenced by: §14.6(ii), §14.9(i) Permalink: http://dlmf.nist.gov/14.9.E4 Encodings: TeX, pMML, png See also: Annotations for §14.9(i), §14.9 and Ch.14

14.9.5	$\displaystyle\mathsf{P}^{\mu}_{-\nu-1}\left(x\right)$	$\displaystyle=\mathsf{P}^{\mu}_{\nu}\left(x\right),$
14.9.5	$\displaystyle\mathsf{P}^{-\mu}_{-\nu-1}\left(x\right)$	$\displaystyle=\mathsf{P}^{-\mu}_{\nu}\left(x\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.2.1 Referenced by: §14.16(i) Permalink: http://dlmf.nist.gov/14.9.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §14.9(i), §14.9 and Ch.14

14.9.6		$\pi\cos\left(\nu\pi\right)\cos\left(\mu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x% \right)=\sin\left((\nu+\mu)\pi\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\sin% \left((\nu-\mu)\pi\right)\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right).$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Permalink: http://dlmf.nist.gov/14.9.E6 Encodings: TeX, pMML, png See also: Annotations for §14.9(i), §14.9 and Ch.14

§14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$

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Keywords:: Ferrers functions, connection formulas
Notes:: See Olver (1997b, p. 188).
Permalink:: http://dlmf.nist.gov/14.9.ii
See also:: Annotations for §14.9 and Ch.14

14.9.7		$\frac{\sin\left((\nu-\mu)\pi\right)}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{% \mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\frac{\sin\left(\mu\pi\right)}{% \Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(-x\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.15(i), §14.15(ii), §14.20(ix) Permalink: http://dlmf.nist.gov/14.9.E7 Encodings: TeX, pMML, png See also: Annotations for §14.9(ii), §14.9 and Ch.14

14.9.8		$\tfrac{1}{2}\pi\sin\left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x% \right)=-\cos\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)-% \mathsf{Q}^{-\mu}_{\nu}\left(-x\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.11, §14.7(iii), §14.8(i) Permalink: http://dlmf.nist.gov/14.9.E8 Encodings: TeX, pMML, png See also: Annotations for §14.9(ii), §14.9 and Ch.14

14.9.9		$\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(% x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(-x\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.15(i), §14.15(ii), §14.15(v) Permalink: http://dlmf.nist.gov/14.9.E9 Encodings: TeX, pMML, png See also: Annotations for §14.9(ii), §14.9 and Ch.14

14.9.10		$(2/\pi)\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\cos% \left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\mathsf{P}^{-% \mu}_{\nu}\left(-x\right).$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.11, §14.15(i), §14.15(ii), §14.15(v), §14.20(i), §14.7(iii), §14.8(i) Permalink: http://dlmf.nist.gov/14.9.E10 Encodings: TeX, pMML, png See also: Annotations for §14.9(ii), §14.9 and Ch.14

§14.9(iii) Connections Between $P^{\pm\mu}_{\nu}\left(x\right)$ , $P^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\boldsymbol{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)$

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Keywords:: associated Legendre functions, connection formulas
Notes:: See Olver (1997b, p. 171).
Permalink:: http://dlmf.nist.gov/14.9.iii
See also:: Annotations for §14.9 and Ch.14

14.9.11	$\displaystyle P^{-\mu}_{-\nu-1}\left(x\right)$	$\displaystyle=P^{-\mu}_{\nu}\left(x\right),$
14.9.11	$\displaystyle P^{\mu}_{-\nu-1}\left(x\right)$	$\displaystyle=P^{\mu}_{\nu}\left(x\right),$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.2.1 Referenced by: §14.16(i), §14.19(v), §14.21(iii) Permalink: http://dlmf.nist.gov/14.9.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §14.9(iii), §14.9 and Ch.14

14.9.12		$\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=-\frac{\boldsymbol{Q}^{\mu% }_{\nu}\left(x\right)}{\Gamma\left(\mu-\nu\right)}+\frac{\boldsymbol{Q}^{\mu}_% {-\nu-1}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.2.2 (modified) Referenced by: §14.20(i), §14.8(ii), §14.8(iii) Permalink: http://dlmf.nist.gov/14.9.E12 Encodings: TeX, pMML, png See also: Annotations for §14.9(iii), §14.9 and Ch.14

14.9.13		$P^{-m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m% +1\right)}P^{m}_{\nu}\left(x\right),$
$\nu\neq m-1,m-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $x$ : real variable, $m$ : nonnegative integer and $\nu$ : general degree Referenced by: §14.19(v), §14.6(ii) Permalink: http://dlmf.nist.gov/14.9.E13 Encodings: TeX, pMML, png See also: Annotations for §14.9(iii), §14.9 and Ch.14

14.9.14		$\boldsymbol{Q}^{-\mu}_{\nu}\left(x\right)=\boldsymbol{Q}^{\mu}_{\nu}\left(x% \right),$
ⓘ Symbols: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.2.6 (modified) Referenced by: §14.23, §14.9(iv) Permalink: http://dlmf.nist.gov/14.9.E14 Encodings: TeX, pMML, png See also: Annotations for §14.9(iii), §14.9 and Ch.14

14.9.15		$\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=% \frac{P^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}-\frac{P^{-\mu% }_{\nu}\left(x\right)}{\Gamma\left(\nu-\mu+1\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.2.5 (modified) Permalink: http://dlmf.nist.gov/14.9.E15 Encodings: TeX, pMML, png See also: Annotations for §14.9(iii), §14.9 and Ch.14

§14.9(iv) Whipple’s Formula

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Keywords:: Whipple’s formula, associated Legendre functions
Notes:: See Olver (1997b, p. 174). For (14.9.17) use (14.9.14) and (14.9.16).
Permalink:: http://dlmf.nist.gov/14.9.iv
See also:: Annotations for §14.9 and Ch.14

14.9.16		$\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(\tfrac{1}{2}\pi\right)^{1/2}% \left(x^{2}-1\right)^{-1/4}\*P^{-\nu-(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1% \right)^{-1/2}\right).$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree A&S Ref: 8.2.7 (modified) Referenced by: §14.19(v), §14.21(iii), §14.9(iv) Permalink: http://dlmf.nist.gov/14.9.E16 Encodings: TeX, pMML, png See also: Annotations for §14.9(iv), §14.9 and Ch.14

Equivalently,

14.9.17		$P^{\mu}_{\nu}\left(x\right)=(2/\pi)^{1/2}\left(x^{2}-1\right)^{-1/4}\*% \boldsymbol{Q}^{\nu+(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1\right)^{-1/2}% \right).$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first 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, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.19(v), §14.9(iv) Permalink: http://dlmf.nist.gov/14.9.E17 Encodings: TeX, pMML, png See also: Annotations for §14.9(iv), §14.9 and Ch.14

14.8 Behavior at Singularities 14.10 Recurrence Relations and Derivatives

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§14.9 Connection Formulas

Contents

§14.9(i) Connections Between 𝖯ν±μ⁡(x), 𝖯−ν−1±μ⁡(x), 𝖰ν±μ⁡(x), 𝖰−ν−1μ⁡(x)

§14.9(ii) Connections Between 𝖯ν±μ⁡(±x), 𝖰ν−μ⁡(±x), 𝖰νμ⁡(x)

§14.9(iii) Connections Between Pν±μ⁡(x), P−ν−1±μ⁡(x), 𝑸ν±μ⁡(x), 𝑸−ν−1μ⁡(x)

§14.9(iv) Whipple’s Formula

§14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$

§14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$ , $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$

§14.9(iii) Connections Between $P^{\pm\mu}_{\nu}\left(x\right)$ , $P^{\pm\mu}_{-\nu-1}\left(x\right)$ , $\boldsymbol{Q}^{\pm\mu}_{\nu}\left(x\right)$ , $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)$