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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.13 Continued Fractions

We use the terminology of §1.12(ii). The following formulae are explicit cases of (18.2.34)–(18.2.36); this area is fully explored in §§18.30(vi) and 18.30(vii).

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 a11x+1212(3x)+2313(5x)+3414(7x)+,

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).