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

§14.14 Continued Fractions

Notes:: (14.14.1) follows from (14.10.6). (14.14.3) follows from (14.10.3), with $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ replaced by $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ . For further details see Gil et al. (2000).
Keywords:: associated Legendre functions, continued fractions
Referenced by:: §14.32
Permalink:: http://dlmf.nist.gov/14.14

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

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

where

14.14.2

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

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

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

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

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

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

where now

14.14.4

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

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

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

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 14.13 Trigonometric Expansions 14.15 Uniform Asymptotic Approximations