§14.9 Connection Formulas

Referenced by:: §14.2(iv)
Permalink:: http://dlmf.nist.gov/14.9

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

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

Notes:: See Olver (1997b, p. 186). For (14.9.3) and (14.9.4) use (14.9.1) and (14.9.2).
Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.9.i

14.9.1 $\frac{\pi\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{2\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)+\frac{\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.9(i)
Permalink:: http://dlmf.nist.gov/14.9.E1
Encodings:: TeX, pMML, png

14.9.2 $\frac{2\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{\pi\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)-\frac{\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.8(i), §14.9(i)
Permalink:: http://dlmf.nist.gov/14.9.E2
Encodings:: TeX, pMML, png

14.9.3 $\mathop{\mathsf{P}^{{-m}}_{{\nu}}\/}\nolimits\!\left(x\right)=(-1)^{m}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}\mathop{\mathsf{P}^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $x$ : 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

14.9.4 $\mathop{\mathsf{Q}^{{-m}}_{{\nu}}\/}\nolimits\!\left(x\right)=(-1)^{m}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}\mathop{\mathsf{Q}^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right),$ $\nu\neq m-1,m-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.9.5

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

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

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.9.6 $\pi\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\sin\/}\nolimits\!\left((\nu+\mu)\pi\right)\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)-\mathop{\sin\/}\nolimits\!\left((\nu-\mu)\pi\right)\mathop{\mathsf{Q}^{{\mu}}_{{-\nu-1}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.9.E6
Encodings:: TeX, pMML, png

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

Notes:: See Olver (1997b, p. 188).
Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.9.ii

14.9.7 $\frac{\mathop{\sin\/}\nolimits\!\left((\nu-\mu)\pi\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)-\frac{\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(-x\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\sin\/}\nolimits z$ : 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

14.9.8 $\tfrac{1}{2}\pi\mathop{\sin\/}\nolimits\!\left((\nu-\mu)\pi\right)\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=-\mathop{\cos\/}\nolimits\!\left((\nu-\mu)\pi\right)\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)-\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(-x\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order 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

14.9.9 $\frac{2}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)}\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=-\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)+\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(-x\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : 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

14.9.10 $(2/\pi)\mathop{\sin\/}\nolimits\!\left((\nu-\mu)\pi\right)\mathop{\mathsf{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\cos\/}\nolimits\!\left((\nu-\mu)\pi\right)\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)-\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(-x\right).$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order 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

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

Notes:: See Olver (1997b, p. 171).
Keywords:: associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.9.iii

14.9.11

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

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

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

14.9.12 $\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)}+\frac{\mathop{\boldsymbol{Q}^{{\mu}}_{{-\nu-1}}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\cos\/}\nolimits z$ : 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

14.9.13 $\mathop{P^{{-m}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1\right)}\mathop{P^{{m}}_{{\nu}}\/}\nolimits\!\left(x\right),$ $\nu\neq m-1,m-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

14.9.14 $\mathop{\boldsymbol{Q}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

14.9.15 $\frac{2\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{\pi}\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}-\frac{\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\sin\/}\nolimits z$ : 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

§14.9(iv) Whipple’s Formula

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

14.9.16 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(\tfrac{1}{2}\pi\right)^{{1/2}}\left(x^{2}-1\right)^{{-1/4}}\*\mathop{P^{{-\nu-(1/2)}}_{{-\mu-(1/2)}}\/}\nolimits\!\left(x\left(x^{2}-1\right)^{{-1/2}}\right).$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $x$ : 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

Equivalently,

14.9.17 $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=(2/\pi)^{{1/2}}\left(x^{2}-1\right)^{{-1/4}}\*\mathop{\boldsymbol{Q}^{{\nu+(1/2)}}_{{-\mu-(1/2)}}\/}\nolimits\!\left(x\left(x^{2}-1\right)^{{-1/2}}\right).$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $x$ : 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

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