Gauss–Laguerre formula
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1: 3.5 Quadrature
2: 6.18 Methods of Computation
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►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998).
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3: 18.5 Explicit Representations
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§18.5(ii) Rodrigues Formulas
… ►Related formula: … ►Laguerre
… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►Laguerre
…4: Bibliography G
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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.
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Asymptotics and bounds for the zeros of Laguerre polynomials: A survey.
J. Comput. Appl. Math. 144 (1-2), pp. 7–27.
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Werke. Band II.
pp. 436–447 (German).
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Construction of Gauss-Christoffel quadrature formulas.
Math. Comp. 22, pp. 251–270.
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Explicit formulas for Bernoulli numbers.
Amer. Math. Monthly 79, pp. 44–51.
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5: 10.74 Methods of Computation
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►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 or is sufficiently small in absolute value.
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►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.
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►For applications of generalized Gauss–Laguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions for and see Gautschi (2002a).
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6: 18.3 Definitions
§18.3 Definitions
►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►As given by a Rodrigues formula (18.5.5).
7: 18.30 Associated OP’s
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►where the generalized hypergeometric function is defined by (16.2.1).
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►For Gauss’ hypergeometric function see (15.2.1).
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§18.30(iii) Associated Laguerre Polynomials
►The recursion relation for the associated Laguerre polynomials, see (18.30.2), (18.30.3) is ►8: 18.17 Integrals
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►For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977).
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►For similar formulas for ultraspherical polynomials see Durand (1975), and for Jacobi and Laguerre polynomials see Durand (1978).
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Laguerre
… ►Laguerre
… ►Laguerre
…9: Bibliography M
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A determinant formula for a class of rational solutions of Painlevé V equation.
Nagoya Math. J. 168, pp. 1–25.
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Applying -Laguerre polynomials to the derivation of -deformed energies of oscillator and Coulomb systems.
Romanian Reports in Physics 57 (1), pp. 25–34.
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A -analog of the Gauss summation theorem for hypergeometric series in
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Adv. in Math. 72 (1), pp. 59–131.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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The -analogue of Stirling’s formula.
Rocky Mountain J. Math. 14 (2), pp. 403–413.
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10: Bibliography T
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Laguerre polynomials: Asymptotics for large degree.
Technical report
Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.
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Asymptotic estimates for Laguerre polynomials.
Z. Angew. Math. Phys. 41 (1), pp. 114–126.
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Large parameter cases of the Gauss hypergeometric function.
J. Comput. Appl. Math. 153 (1-2), pp. 441–462.
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Is Gauss quadrature better than Clenshaw-Curtis?.
SIAM Rev. 50 (1), pp. 67–87.
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Sul comportamento asintotico dell’-esimo polinomio di Laguerre nell’intorno dell’ascissa
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Comment. Math. Helv. 22, pp. 150–167.
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