# §14.14 Continued Fractions

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

where

 14.14.2 $\displaystyle x_{k}$ $\displaystyle=\tfrac{1}{4}(\nu-\mu-k+1)(\nu+\mu+k)\left(x^{2}-1\right),$ $\displaystyle y_{k}$ $\displaystyle=(\mu+k)x,$ ⓘ 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 $(\nu-\mu)\frac{Q^{\mu}_{\nu}\left(x\right)}{Q^{\mu}_{\nu-1}\left(x\right)}=% \cfrac{x_{0}}{y_{0}-\cfrac{x_{1}}{y_{1}-\cfrac{x_{2}}{y_{2}-\cdots}}},$ $\nu\neq\mu$,

where now

 14.14.4 $\displaystyle x_{k}$ $\displaystyle=(\nu+\mu+k)(\nu-\mu+k),$ $\displaystyle y_{k}$ $\displaystyle=(2\nu+2k+1)x,$ ⓘ 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 $k=0,1,2,\dots$.