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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
2: Bibliography
  • M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
  • A. M. Al-Rashed and N. Zaheer (1985) Zeros of Stieltjes and Van Vleck polynomials and applications. J. Math. Anal. Appl. 110 (2), pp. 327–339.
  • W. A. Al-Salam (1990) Characterization theorems for orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.
  • M. Alam (1979) Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.
  • R. Askey (1989) Continuous q -Hermite Polynomials when q > 1 . In q -series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 18, pp. 151–158.
  • 3: Joris Van der Jeugt
    Profile
    Joris Van der Jeugt
    Joris Van der Jeugt (b. … His results for Lie superalgebra representations continue to inspire many scientists. … In September 2024, Van der Jeugt was named Associate Editor for DLMF Chapter …
    4: 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).
    5: 31.15 Stieltjes Polynomials
    The V ( z ) are called Van Vleck polynomials and the corresponding S ( z ) Stieltjes polynomials. …
    31.15.2 j = 1 N γ j / 2 z k a j + j = 1 j k n 1 z k z j = 0 , k = 1 , 2 , , n .
    If t k is a zero of the Van Vleck polynomial V ( z ) , corresponding to an n th degree Stieltjes polynomial S ( z ) , and z 1 , z 2 , , z n 1 are the zeros of S ( z ) (the derivative of S ( z ) ), then t k is either a zero of S ( z ) or a solution of the equation … 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. …
    6: 30.10 Series and Integrals
    For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …
    7: 19.35 Other Applications
    §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 π to high precision (Borwein and Borwein (1987, p. 26)). …
    8: 5.21 Methods of Computation
    For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). … For a comprehensive survey see van der Laan and Temme (1984, Chapter III). …
    9: 7.22 Methods of Computation
    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. … For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. V).
    10: 10.74 Methods of Computation
    §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). …