14 Legendre and Related FunctionsReal Arguments14.13 Trigonometric Expansions14.15 Uniform Asymptotic Approximations

§14.14 Continued Fractions

ⓘ

Keywords:

associated Legendre functions, continued fractions

Notes:

(14.14.1) follows from (14.10.6). (14.14.3) follows from (14.10.3), with ${𝖯}_{\nu }^{\mu }\left(x\right)$ replaced by $Q_{\nu }^{\mu }\left(x\right)$. 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		$$\frac{1}{2}{\left(x^2-1\right)}^{1/2}\frac{P_{\nu }^{\mu }\left(x\right)}{P_{\nu }^{\mu -1}\left(x\right)}=\frac{x_0}{y_0+}\frac{x_1}{y_1+}\frac{x_2}{y_2+}\mathrm{\cdots },$$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: 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		$x_k$	$=\frac{1}{4}(\nu -\mu -k+1)(\nu +\mu +k)\left(x^2-1\right),$
		$y_k$	$=(\mu +k)x,$
ⓘ Symbols: $x$: 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 $k=0,1,2,\mathrm{\dots }$.

14.14.3		$$(\nu -\mu )\frac{Q_{\nu }^{\mu }\left(x\right)}{Q_{\nu -1}^{\mu }\left(x\right)}=\frac{x_0}{y_0-}\frac{x_1}{y_1-}\frac{x_2}{y_2-}\mathrm{\cdots },$$
$\nu \ne \mu$,
ⓘ Symbols: $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $x$: 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		$x_k$	$=(\nu +\mu +k)(\nu -\mu +k),$
		$y_k$	$=(2\nu +2k+1)x,$
ⓘ Symbols: $x$: 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 $k=0,1,2,\mathrm{\dots }$.

14.13 Trigonometric Expansions14.15 Uniform Asymptotic Approximations

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