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Van Vleck theorem

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1: 1.12 Continued Fractions
Pringsheim’s Theorem
Van Vleck’s Theorem
2: 30.10 Series and Integrals
For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …
3: 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.
  • 4: 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)). …
    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: 28.27 Addition Theorems
    §28.27 Addition Theorems
    Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
    7: Bibliography V
  • A. L. Van Buren, R. V. Baier, S. Hanish, and B. J. King (1972) Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.
  • A. L. Van Buren and J. E. Boisvert (2007) Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.
  • Van Buren (website) Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation
  • H. C. van de Hulst (1957) Light Scattering by Small Particles. John Wiley and Sons. Inc., New York.
  • H. C. van de Hulst (1980) Multiple Light Scattering. Vol. 1, Academic Press, New York.
  • 8: 25.6 Integer Arguments
    25.6.4 ζ ( 2 n ) = 0 , n = 1 , 2 , 3 , .
    25.6.8 ζ ( 2 ) = 3 k = 1 1 k 2 ( 2 k k ) .
    25.6.9 ζ ( 3 ) = 5 2 k = 1 ( 1 ) k 1 k 3 ( 2 k k ) .
    25.6.10 ζ ( 4 ) = 36 17 k = 1 1 k 4 ( 2 k k ) .
    9: 6.17 Physical Applications
    For applications in astrophysics, see also van de Hulst (1980). …
    10: Bibliography D
  • N. G. de Bruijn (1937) Integralen voor de ζ -functie van Riemann. Mathematica (Zutphen) B5, pp. 170–180 (Dutch).
  • B. Deconinck, M. Heil, A. Bobenko, M. van Hoeij, and M. Schmies (2004) Computing Riemann theta functions. Math. Comp. 73 (247), pp. 1417–1442.
  • B. Deconinck and M. van Hoeij (2001) Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.
  • J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
  • T. M. Dunster (1999) Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.