homogeneous harmonic polynomials
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1: 14.30 Spherical and Spheroidal Harmonics
§14.30 Spherical and Spheroidal Harmonics
… ► … ►In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See Andrews et al. (1999, Chapter 9). … ►2: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
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31.5.2
►is a polynomial of degree , and hence a solution of (31.2.1) that is analytic at all three finite singularities .
These solutions are the Heun polynomials.
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3: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
… ►Normalization
… ►Orthogonal Invariance
… ►Summation
… ►Mean-Value
…4: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …5: 18.3 Definitions
§18.3 Definitions
… ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to for these polynomials and for coefficients up to for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►Bessel polynomials
►Bessel polynomials are often included among the classical OP’s. …6: Bibliography G
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A harmonic mean inequality for the gamma function.
SIAM J. Math. Anal. 5 (2), pp. 278–281.
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Evaluation of toroidal harmonics.
Comput. Phys. Comm. 124 (1), pp. 104–122.
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Computing the zeros and turning points of solutions of second order homogeneous linear ODEs.
SIAM J. Numer. Anal. 41 (3), pp. 827–855.
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Analysis in homogeneous domains.
Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).
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Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials.
J. Phys. A 47 (1), pp. 015203, 26 pp..
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7: 17.17 Physical Applications
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►See Kassel (1995).
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►It involves -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials.
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8: 12.17 Physical Applications
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►Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator.
►Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs.
For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978).
►Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).
9: Donald St. P. Richards
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►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability.
He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T.
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10: Bibliography F
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Theory and Computation of Spheroidal Harmonics with General Arguments.
Master’s Thesis, The University of Western Australia, Department of Physics.
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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II.
J. Math. Anal. Appl. 7 (3), pp. 440–451.
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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order.
J. Math. Anal. Appl. 6 (3), pp. 394–403.
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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III.
J. Math. Anal. Appl. 12 (3), pp. 593–601.
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Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen.
Math. Ann. 63 (3), pp. 301–321.
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