associated Laguerre functions
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1: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►The functions
defined by (33.14.14) are the hydrogenic bound states in attractive Coulomb potentials; their polynomial components are often called associated Laguerre functions; see Christy and Duck (1961) and Bethe and Salpeter (1977).
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2: 18.39 Applications in the Physical Sciences
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d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s
… ►(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials . …3: 18.30 Associated OP’s
4: Bibliography
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Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions.
J. Math. Physics 33, pp. 111–116.
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SPHEREPACK 2.0: A Model Development Facility.
NCAR Technical Note
Technical Report TN-436-STR, National Center for Atmospheric Research.
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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions.
J. Heat Transfer 118 (3), pp. 789–792.
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Associated Laguerre and Hermite polynomials.
Proc. Roy. Soc. Edinburgh 96A, pp. 15–37.
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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.
J. Math. Anal. Appl. 416 (1), pp. 52–80.
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5: Errata
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Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions
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Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, and the Laguerre polynomial, , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23
6: Bibliography D
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Strong and ratio asymptotics for Laguerre polynomials revisited.
J. Math. Anal. Appl. 403 (2), pp. 477–486.
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Differential-difference operators associated to reflection groups.
Trans. Amer. Math. Soc. 311 (1), pp. 167–183.
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A Laguerre polynomial orthogonality and the hydrogen atom.
Anal. Appl. (Singap.) 1 (2), pp. 177–188.
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Uniform asymptotic expansions for associated Legendre functions of large order.
Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.
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Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions.
Stud. Appl. Math. 113 (3), pp. 245–270.
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7: 10.74 Methods of Computation
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►Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin 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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§10.74(vi) Zeros and Associated Values
…8: 18.3 Definitions
§18.3 Definitions
►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of for Jacobi polynomials, in powers of for the other cases). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …9: Bibliography L
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Some differential equations and associated integral equations.
Quart. J. Math. (Oxford) 5, pp. 81–97.
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Orthogonal polynomials, duality and association schemes.
SIAM J. Math. Anal. 13 (4), pp. 656–663.
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Co-recursive associated Jacobi polynomials.
J. Comput. Appl. Math. 57 (1-2), pp. 203–213.
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Polylogarithms and Associated Functions.
North-Holland Publishing Co., New York.
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The spectral analysis of three families of exceptional Laguerre polynomials.
J. Approx. Theory 202, pp. 5–41.
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