§18.13 Continued Fractions

Notes:: See Lorentzen and Waadeland (1992, pp. 446–448).
Keywords:: classical orthogonal polynomials, continued fractions
Permalink:: http://dlmf.nist.gov/18.13

We use the terminology of §1.12(ii).

¶ Chebyshev

Keywords:: Chebyshev polynomials, continued fractions

$\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ 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

and $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ 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

¶ Legendre

Keywords:: Legendre polynomials

$\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ 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

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

¶ Laguerre

Keywords:: Laguerre polynomials

$\mathop{L_{{n}}\/}\nolimits\!\left(x\right)$ 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

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

¶ Hermite

Keywords:: Hermite polynomials, classical orthogonal polynomials, continued fractions

$\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ 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 Cuyt et al. (2008, pp. 91–99).