14 Legendre and Related Functions Real Arguments 14.13 Trigonometric Expansions 14.15 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 $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ 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		$\tfrac{1}{2}\left(x^{2}-1\right)^{1/2}\frac{P^{\mu}_{\nu}\left(x\right)}{P^{% \mu-1}_{\nu}\left(x\right)}=\cfrac{x_{0}}{y_{0}+\cfrac{x_{1}}{y_{1}+\cfrac{x_{% 2}}{y_{2}+\cdots}}},$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$\displaystyle x_{k}$	$\displaystyle=\tfrac{1}{4}(\nu-\mu-k+1)(\nu+\mu+k)\left(x^{2}-1\right),$
14.14.2		$\displaystyle y_{k}$	$\displaystyle=(\mu+k)x,$
ⓘ 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 $k=0,1,2,\dots$ .

14.14.3		$(\nu-\mu)\frac{Q^{\mu}_{\nu}\left(x\right)}{Q^{\mu}_{\nu-1}\left(x\right)}=% \cfrac{x_{0}}{y_{0}-\cfrac{x_{1}}{y_{1}-\cfrac{x_{2}}{y_{2}-\cdots}}},$
$\nu\neq\mu$ ,
ⓘ Symbols: $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$\displaystyle x_{k}$	$\displaystyle=(\nu+\mu+k)(\nu-\mu+k),$
14.14.4		$\displaystyle y_{k}$	$\displaystyle=(2\nu+2k+1)x,$
ⓘ 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 $k=0,1,2,\dots$ .

14.13 Trigonometric Expansions 14.15 Uniform Asymptotic Approximations

© 2010–2026 NIST / Disclaimer / Feedback; Version 1.2.8; Release date 2026-09-15.

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov