# §18.13 Continued Fractions

We use the terminology of §1.12(ii).

## Chebyshev

$T_{n}\left(x\right)$ is the denominator of the $n$th approximant to:

 18.13.1 $\cfrac{-1}{x+\cfrac{-1}{2x+\cfrac{-1}{2x+}}}\cdots,$ ⓘ Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/18.13.E1 Encodings: TeX, pMML, png See also: Annotations for 18.13, 18.13 and 18

and $U_{n}\left(x\right)$ is the denominator of the $n$th approximant to:

 18.13.2 $\cfrac{-1}{2x+\cfrac{-1}{2x+\cfrac{-1}{2x+}}}\cdots.$ ⓘ Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/18.13.E2 Encodings: TeX, pMML, png See also: Annotations for 18.13, 18.13 and 18

## Legendre

$P_{n}\left(x\right)$ is the denominator of the $n$th approximant to:

 18.13.3 $\cfrac{a_{1}}{x+\cfrac{-\frac{1}{2}}{\frac{3}{2}x+\cfrac{-\frac{2}{3}}{\frac{5% }{3}x+\cfrac{-\frac{3}{4}}{\frac{7}{4}x+}}}}\cdots,$ ⓘ Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/18.13.E3 Encodings: TeX, pMML, png See also: Annotations for 18.13, 18.13 and 18

where $a_{1}$ is an arbitrary nonzero constant.

## Laguerre

$L_{n}\left(x\right)$ is the denominator of the $n$th approximant to:

 18.13.4 $\cfrac{a_{1}}{1-x+\cfrac{-\frac{1}{2}}{\frac{1}{2}(3-x)+\cfrac{-\frac{2}{3}}{% \frac{1}{3}(5-x)+\cfrac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}}}}\cdots,$ ⓘ Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/18.13.E4 Encodings: TeX, pMML, png See also: Annotations for 18.13, 18.13 and 18

where $a_{1}$ is again an arbitrary nonzero constant.

## Hermite

$H_{n}\left(x\right)$ is the denominator of the $n$th approximant to:

 18.13.5 $\cfrac{1}{2x+\cfrac{-2}{2x+\cfrac{-4}{2x+\cfrac{-6}{2x+}}}}\cdots.$ ⓘ Symbols: $x$: real variable Permalink: http://dlmf.nist.gov/18.13.E5 Encodings: TeX, pMML, png See also: Annotations for 18.13, 18.13 and 18