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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.13 Continued Fractions

We use the terminology of §1.12(ii).

Chebyshev

Tn(x) is the denominator of the nth approximant to:

18.13.1 -1x+-12x+-12x+,

and Un(x) is the denominator of the nth approximant to:

18.13.2 -12x+-12x+-12x+.

Legendre

Pn(x) is the denominator of the nth approximant to:

18.13.3 a1x+-1232x+-2353x+-3474x+,

where a1 is an arbitrary nonzero constant.

Laguerre

Ln(x) is the denominator of the nth approximant to:

18.13.4 a11-x+-1212(3-x)+-2313(5-x)+-3414(7-x)+,

where a1 is again an arbitrary nonzero constant.

Hermite

Hn(x) is the denominator of the nth approximant to:

18.13.5 12x+-22x+-42x+-62x+.

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