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Christoffel coefficients (or numbers)

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1: 3.5 Quadrature
3.5.18 w k = a b p n ( x ) ( x - x k ) p n ( x k ) w ( x ) d x .
The w k are also known as Christoffel coefficients or Christoffel numbers and they are all positive. The remainder is given by …
2: Bibliography N
  • National Bureau of Standards (1944) Tables of Lagrangian Interpolation Coefficients. Columbia University Press, New York.
  • National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.
  • P. Nevai (1986) Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.
  • 3: Errata
  • Section 34.1

    The relation between Clebsch-Gordan and 3 j symbols was clarified, and the sign of m 3 was changed for readability. The reference Condon and Shortley (1935) for the Clebsch-Gordan coefficients was replaced by Edmonds (1974) and Rotenberg et al. (1959) and the references for 3 j , 6 j , 9 j symbols were made more precise in §34.1.

  • Section 34.1

    The reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for 3 j , 6 j , 9 j symbols were made more precise.

  • Subsection 1.2(i)

    A sentence was added after (1.2.1) to refer to (1.2.6) as the definition of the binomial coefficient ( z k ) when z is complex. As a notational clarification, wherever n appeared originally in (1.2.6)–(1.2.9), it was replaced by z .

  • Subsection 14.18(iii)

    This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

  • Equation (10.20.14)
    10.20.14 B 3 ( 0 ) = - 959 71711 84603 25 47666 37125 00000 2 1 3

    Originally this coefficient was given incorrectly as B 3 ( 0 ) = - 430 99056 39368 59253 5 68167 34399 42500 00000 2 1 3 . The other coefficients in this equation have not been changed.

    Reported 2012-05-11 by Antony Lee.