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 $\frac{\pi\sin\left(\mu\pi\right)}{2\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-% \mu}_{\nu}\left(x\right)=-\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)+\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right).$

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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, : 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 $\frac{2\sin\left(\mu\pi\right)}{\pi\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-% \mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{\mu% }_{\nu}\left(x\right)-\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right),$

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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, : 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 $\mathsf{P}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{P}^{m}_{\nu}\left(x\right),$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable, : 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 $\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}\left(x\right),$ $\nu\neq m-1,m-2,\dots$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable, : 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

$\mathsf{P}^{\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{\mu}_{\nu}\left(x\right),$

$\mathsf{P}^{-\mu}_{-\nu-1}\left(x\right)=\mathsf{P}^{-\mu}_{\nu}\left(x\right),$

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : 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 $\pi\cos\left(\nu\pi\right)\cos\left(\mu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x% \right)=\sin\left((\nu+\mu)\pi\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\sin% \left((\nu-\mu)\pi\right)\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right).$

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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, : 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 $\frac{\sin\left((\nu-\mu)\pi\right)}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{% \mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\frac{\sin\left(\mu\pi\right)}{% \Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(-x\right),$

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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, : 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 $\tfrac{1}{2}\pi\sin\left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x% \right)=-\cos\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)-% \mathsf{Q}^{-\mu}_{\nu}\left(-x\right),$

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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, : 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 $\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(% x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(-x\right),$

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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, : 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 $(2/\pi)\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\cos% \left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\mathsf{P}^{-% \mu}_{\nu}\left(-x\right).$

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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, : 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

$P^{-\mu}_{-\nu-1}\left(x\right)=P^{-\mu}_{\nu}\left(x\right),$

$P^{\mu}_{-\nu-1}\left(x\right)=P^{\mu}_{\nu}\left(x\right),$

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : 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 $\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=-\frac{\boldsymbol{Q}^{\mu% }_{\nu}\left(x\right)}{\Gamma\left(\mu-\nu\right)}+\frac{\boldsymbol{Q}^{\mu}_% {-\nu-1}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}.$

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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, : 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 $P^{-m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m% +1\right)}P^{m}_{\nu}\left(x\right),$ $\nu\neq m-1,m-2,\dots$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : real variable, : 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 $\boldsymbol{Q}^{-\mu}_{\nu}\left(x\right)=\boldsymbol{Q}^{\mu}_{\nu}\left(x% \right),$

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Symbols:: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, : 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 $\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=% \frac{P^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}-\frac{P^{-\mu% }_{\nu}\left(x\right)}{\Gamma\left(\nu-\mu+1\right)}.$

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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, : 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 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(\tfrac{1}{2}\pi\right)^{1/2}% \left(x^{2}-1\right)^{-1/4}\*P^{-\nu-(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1% \right)^{-1/2}\right).$

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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, : 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 $P^{\mu}_{\nu}\left(x\right)=(2/\pi)^{1/2}\left(x^{2}-1\right)^{-1/4}\*% \boldsymbol{Q}^{\nu+(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1\right)^{-1/2}% \right).$

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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, : 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 , , ,

§14.9(ii) Connections Between , ,

§14.9(iii) Connections Between , , ,

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