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Gauss–Laguerre formula

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1: 3.5 Quadrature
GaussLaguerre Formula
Table 3.5.6: Nodes and weights for the 5-point GaussLaguerre formula.
x k w k
Table 3.5.7: Nodes and weights for the 10-point GaussLaguerre formula.
x k w k
Table 3.5.8: Nodes and weights for the 15-point GaussLaguerre formula.
x k w k
Table 3.5.9: Nodes and weights for the 20-point GaussLaguerre formula.
x k w k
2: 6.18 Methods of Computation
For example, the Gauss-Laguerre formula3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). …
3: 18.5 Explicit Representations
§18.5(ii) Rodrigues Formulas
Related formula: …
Laguerre
For corresponding formulas for Chebyshev, Legendre, and the Hermite He n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
Laguerre
4: Bibliography G
  • B. Gabutti and B. Minetti (1981) A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function. J. Comput. Phys. 42 (2), pp. 277–287.
  • L. Gatteschi (2002) Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math. 144 (1-2), pp. 7–27.
  • C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).
  • W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • 5: 10.74 Methods of Computation
    The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument x or z is sufficiently small in absolute value. … In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. … For applications of generalized GaussLaguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions K ν ( z ) for 0 < ν < 1 and 0 < x < see Gautschi (2002a). …
    6: Bibliography M
  • T. Masuda, Y. Ohta, and K. Kajiwara (2002) A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.
  • C. Micu and E. Papp (2005) Applying q -Laguerre polynomials to the derivation of q -deformed energies of oscillator and Coulomb systems. Romanian Reports in Physics 57 (1), pp. 25–34.
  • S. C. Milne (1988) A q -analog of the Gauss summation theorem for hypergeometric series in U ( n ) . Adv. in Math. 72 (1), pp. 59–131.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • D. S. Moak (1984) The q -analogue of Stirling’s formula. Rocky Mountain J. Math. 14 (2), pp. 403–413.
  • 7: Bibliography T
  • N. M. Temme (1986) Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.
  • N. M. Temme (1990a) Asymptotic estimates for Laguerre polynomials. Z. Angew. Math. Phys. 41 (1), pp. 114–126.
  • N. M. Temme (2003) Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.
  • L. N. Trefethen (2008) Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.
  • F. G. Tricomi (1949) Sul comportamento asintotico dell’ n -esimo polinomio di Laguerre nell’intorno dell’ascissa 4 n . Comment. Math. Helv. 22, pp. 150–167.
  • 8: Errata
  • Changes


    • In Paragraph Inversion Formula in §35.2, the wording was changed to make the integration variable more apparent.

    • In many cases, the links from mathematical symbols to their definitions were corrected or improved. These links were also enhanced with ‘tooltip’ feedback, where supported by the user’s browser.

  • Other Changes


  • Other Changes


    • To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

      8.12.5 e ± π i a 2 i sin ( π a ) Q ( - a , z e ± π i ) = ± 1 2 erfc ( ± i η a / 2 ) - i T ( a , η )
    • Following a suggestion from James McTavish on 2017-04-06, the recurrence relation u k = ( 6 k - 5 ) ( 6 k - 3 ) ( 6 k - 1 ) ( 2 k - 1 ) 216 k u k - 1 was added to Equation (9.7.2).

    • In §15.2(ii), the unnumbered equation

      lim c - n F ( a , b ; c ; z ) Γ ( c ) = F ( a , b ; - n ; z ) = ( a ) n + 1 ( b ) n + 1 ( n + 1 ) ! z n + 1 F ( a + n + 1 , b + n + 1 ; n + 2 ; z ) , n = 0 , 1 , 2 ,

      was added in the second paragraph. An equation number will be assigned in an expanded numbering scheme that is under current development. Additionally, the discussion following (15.2.6) was expanded.

    • In §15.4(i), due to a report by Louis Klauder on 2017-01-01, and in §15.4(iii), sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula F ( a , b ; a ; z ) = ( 1 - z ) - b .

    • A bibliographic citation was added in §11.13(i).

  • 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.