Van Vleck theorem for continued fractions
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1: 1.12 Continued Fractions
§1.12 Continued Fractions
… ►Fractional Transformations
… ►Pringsheim’s Theorem
… ►Van Vleck’s Theorem
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Asymptotic expansions of spheroidal wave functions.
J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
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Zeros of Stieltjes and Van Vleck polynomials and applications.
J. Math. Anal. Appl. 110 (2), pp. 327–339.
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Zeros of Stieltjes and Van Vleck polynomials.
Trans. Amer. Math. Soc. 252, pp. 197–204.
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Recurrence relations, continued fractions, and orthogonal polynomials.
Mem. Amer. Math. Soc. 49 (300), pp. iv+108.
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Continuous -Hermite Polynomials when
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In
-series and Partitions (Minneapolis, MN, 1988),
IMA Vol. Math. Appl., Vol. 18, pp. 151–158.
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3: 5.10 Continued Fractions
§5.10 Continued Fractions
… ►Also see Cuyt et al. (2008, pp. 223–228), Jones and Thron (1980, pp. 348–350), Lorentzen and Waadeland (1992, pp. 221–224), and Jones and Van Assche (1998).4: 31.15 Stieltjes Polynomials
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►The are called Van Vleck polynomials and the corresponding
Stieltjes polynomials.
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31.15.2
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►If is a zero of the Van Vleck polynomial , corresponding to an th degree Stieltjes polynomial , and are the zeros of (the derivative of ), then is either a zero of or a solution of the equation
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►See Marden (1966), Alam (1979), and Al-Rashed and Zaheer (1985) for further results on the location of the zeros of Stieltjes and Van Vleck polynomials.
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5: 30.10 Series and Integrals
6: 19.35 Other Applications
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§19.35(i) Mathematical
►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute to high precision (Borwein and Borwein (1987, p. 26)). …7: 5.21 Methods of Computation
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►For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3).
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►For a comprehensive survey see van der Laan and Temme (1984, Chapter III).
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8: 7.22 Methods of Computation
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►Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions.
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►For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. V).
9: 10.74 Methods of Computation
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§10.74(v) Continued Fractions
►For applications of the continued-fraction expansions (10.10.1), (10.10.2), (10.33.1), and (10.33.2) to the computation of Bessel functions and modified Bessel functions see Gargantini and Henrici (1967), Amos (1974), Gautschi and Slavik (1978), Tretter and Walster (1980), Thompson and Barnett (1986), and Cuyt et al. (2008). … ►To ensure that no zeros are overlooked, standard tools are the phase principle and Rouché’s theorem; see §1.10(iv). … ►For infinite integrals involving products of Bessel functions of the first kind, see Linz and Kropp (1973), Gabutti (1980), Ikonomou et al. (1995), Lucas (1995), and Van Deun and Cools (2008). …10: 13.5 Continued Fractions
§13.5 Continued Fractions
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13.5.1
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►This continued fraction converges to the meromorphic function of on the left-hand side everywhere in .
For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).
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►This continued fraction converges to the meromorphic function of on the left-hand side throughout the sector .
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