14 Legendre and Related Functions Real Arguments 14.13 Trigonometric Expansions 14.15 Uniform Asymptotic Approximations

§14.14 Continued Fractions

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Keywords:: associated Legendre functions, continued fractions
Notes:: (14.14.1) follows from (14.10.6). (14.14.3) follows from (14.10.3), with $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ replaced by $Q^{\mu}_{\nu}\left(x\right)$ . For further details see Gil et al. (2000).
Referenced by:: 4th item
Permalink:: http://dlmf.nist.gov/14.14
See also:: Annotations for Ch.14

14.14.1 $\tfrac{1}{2}\left(x^{2}-1\right)^{1/2}\frac{P^{\mu}_{\nu}\left(x\right)}{P^{% \mu-1}_{\nu}\left(x\right)}=\cfrac{x_{0}}{y_{0}+\cfrac{x_{1}}{y_{1}+\cfrac{x_{% 2}}{y_{2}+\cdots}}},$

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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, $\nu$ : general degree, $x_{k}$ : numerator and $x_{k}$ : denominator
Referenced by:: §14.14
Permalink:: http://dlmf.nist.gov/14.14.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §14.14 and Ch.14

where

14.14.2

$x_{k}=\tfrac{1}{4}(\nu-\mu-k+1)(\nu+\mu+k)\left(x^{2}-1\right),$

$y_{k}=(\mu+k)x,$

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Symbols:: : real variable, $\mu$ : general order, $\nu$ : general degree, $x_{k}$ : numerator and $x_{k}$ : denominator
Permalink:: http://dlmf.nist.gov/14.14.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §14.14 and Ch.14

provided that $x_{k+1}$ and $y_{k}$ do not vanish simultaneously for any $k=0,1,2,\dots$ .

14.14.3 $(\nu-\mu)\frac{Q^{\mu}_{\nu}\left(x\right)}{Q^{\mu}_{\nu-1}\left(x\right)}=% \cfrac{x_{0}}{y_{0}-\cfrac{x_{1}}{y_{1}-\cfrac{x_{2}}{y_{2}-\cdots}}},$ $\nu\neq\mu$ ,

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Symbols:: $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, : real variable, $\mu$ : general order, $\nu$ : general degree, $x_{k}$ : numerator and $x_{k}$ : denominator
Referenced by:: §14.14
Permalink:: http://dlmf.nist.gov/14.14.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §14.14 and Ch.14

where now

14.14.4

$x_{k}=(\nu+\mu+k)(\nu-\mu+k),$

$y_{k}=(2\nu+2k+1)x,$

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Symbols:: : real variable, $\mu$ : general order, $\nu$ : general degree, $x_{k}$ : numerator and $x_{k}$ : denominator
Permalink:: http://dlmf.nist.gov/14.14.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §14.14 and Ch.14

again provided $x_{k+1}$ and $y_{k}$ do not vanish simultaneously for any $k=0,1,2,\dots$ .

14.13 Trigonometric Expansions 14.15 Uniform Asymptotic Approximations

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