Van Vleck theorem
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1: 1.12 Continued Fractions
2: 30.10 Series and Integrals
3: 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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4: Bibliography
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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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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.
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Zeros of Stieltjes and Van Vleck polynomials.
Trans. Amer. Math. Soc. 252, pp. 197–204.
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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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5: 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)). …6: Joris Van der Jeugt
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Joris Van der Jeugt
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►Joris Van der Jeugt (b.
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►In September 2024, Van der Jeugt was named Associate Editor for DLMF Chapter
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7: Bibliography V
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Calculation of spheroidal wave functions.
J. Acoust. Soc. Amer. 51, pp. 414–416.
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Accurate calculation of the modified Mathieu functions of integer order.
Quart. Appl. Math. 65 (1), pp. 1–23.
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Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation
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Light Scattering by Small Particles.
John Wiley and Sons. Inc., New York.
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Multiple Light Scattering.
Vol. 1, Academic Press, New York.
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