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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: Errata
  • Changes


    • There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

      Transform New Abbreviated Old
      Notation Notation Notation
      Fourier ( f ) ( x ) f ( x )
      Fourier Cosine c ( f ) ( x ) c f ( x )
      Fourier Sine s ( f ) ( x ) s f ( x )
      Laplace ( f ) ( s ) f ( s ) ( f ( t ) ; s )
      Mellin ( f ) ( s ) f ( s ) ( f ; s )
      Hilbert ( f ) ( s ) f ( s ) ( f ; s )
      Stieltjes 𝒮 ( f ) ( s ) 𝒮 f ( s ) 𝒮 ( f ; s )

      Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

    • Several changes have been made in §1.16(vii) to

      1. (i)

        make consistent use of the Fourier transform notations ( f ) , ( ϕ ) and ( u ) where f is a function of one real variable, ϕ is a test function of n variables associated with tempered distributions, and u is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

      2. (ii)

        introduce the partial differential operator D in (1.16.30);

      3. (iii)

        clarify the definition (1.16.32) of the partial differential operator P ( D ) ; and

      4. (iv)

        clarify the use of P ( D ) and P ( x ) in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

    • An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.

    • The validity constraint | ph z | < 1 6 π was added to (9.5.6). Additionally, specific source citations are now given in the metadata for all equations in Chapter 9 Airy and Related Functions.

    • 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.

    • The website’s icons and graphical decorations were upgraded to use SVG, and additional icons and mouse-cursors were employed to improve usability of the interactive figures.

  • Other Changes


  • Other Changes


    • Equations (4.45.8) and (4.45.9) have been replaced with equations that are better for numerically computing arctan x .

    • A new Subsection 13.29(v) Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

    • A new Subsection 14.5(vi) Addendum to §14.5(ii) μ = 0 , ν = 2 , containing the values of Legendre and Ferrers functions for degree ν = 2 has been added.

    • Subsection 14.18(iii) has been altered to identify Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

    • A new Subsection 15.19(v) Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

    • Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

    • Cross-references have been added in §§1.2(i), 10.19(iii), 10.23(ii), 17.2(iii), 18.15(iii), 19.2(iv), 19.16(i).

    • Several small revisions have been made. For details see §§5.11(ii), 10.12, 10.19(ii), 18.9(i), 18.16(iv), 19.7(ii), 22.2, 32.11(v), 32.13(ii).

    • Entries for the Sage computational system have been updated in the Software Index.

    • The default document format for DLMF is now HTML5 which includes MathML providing better accessibility and display of mathematics.

    • All interactive 3D graphics on the DLMF website have been recast using WebGL and X3DOM, improving portability and performance; WebGL it is now the default format.

  • 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.

  • Table 18.9.1


    The coefficient A n for C n ( λ ) ( x ) in the first row of this table originally omitted the parentheses and was given as 2 n + λ n + 1 , instead of 2 ( n + λ ) n + 1 .

    p n ( x ) A n B n C n
    C n ( λ ) ( x ) 2 ( n + λ ) n + 1 0 n + 2 λ - 1 n + 1

    Reported 2010-09-16 by Kendall Atkinson.

  • 3: 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.