§18.13 Continued Fractions

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

18.13.1		$$\frac{-1}{x+}\frac{-1}{2x+}\frac{-1}{2x+}\mathrm{\cdots },$$
ⓘ 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 Ch.18

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

18.13.2		$$\frac{-1}{2x+}\frac{-1}{2x+}\frac{-1}{2x+}\mathrm{\cdots }.$$
ⓘ 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 Ch.18

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

18.13.3		$$\frac{a_1}{x+}\frac{-\frac{1}{2}}{\frac{3}{2}x+}\frac{-\frac{2}{3}}{\frac{5}{3}x+}\frac{-\frac{3}{4}}{\frac{7}{4}x+}\mathrm{\cdots },$$
ⓘ 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 Ch.18

where $a_1$ is an arbitrary nonzero constant.

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

18.13.4		$$\frac{a_1}{1-x+}\frac{-\frac{1}{2}}{\frac{1}{2}(3-x)+}\frac{-\frac{2}{3}}{\frac{1}{3}(5-x)+}\frac{-\frac{3}{4}}{\frac{1}{4}(7-x)+}\mathrm{\cdots },$$
ⓘ 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 Ch.18

where $a_1$ is again an arbitrary nonzero constant.

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

18.13.5		$$\frac{1}{2x+}\frac{-2}{2x+}\frac{-4}{2x+}\frac{-6}{2x+}\mathrm{\cdots }.$$
ⓘ 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 Ch.18

See also Cuyt et al. (2008, pp. 91–99).