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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
  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Usability

    Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

  • Chapter 35 Functions of Matrix Argument

    The generalized hypergeometric function of matrix argument F q p ( a 1 , , a p ; b 1 , , b q ; T ) , was linked inadvertently as its single variable counterpart F q p ( a 1 , , a p ; b 1 , , b q ; T ) . Furthermore, the Jacobi function of matrix argument P ν ( γ , δ ) ( T ) , and the Laguerre function of matrix argument L ν ( γ ) ( T ) , were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by P ν ( γ , δ ) ( T ) , and L ν ( γ ) ( T ) . In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

  • Subsections 15.4(i), 15.4(ii)

    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 .

    Report by Louis Klauder on 2017-01-01.

  • Equation (18.15.22)

    Because of the use of the O order symbol on the right-hand side, the asymptotic expansion for the generalized Laguerre polynomial L n ( α ) ( ν x ) was rewritten as an equality.