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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
31.5.2 Hp n , m ( a , q n , m ; - n , β , γ , δ ; z ) = H ( a , q n , m ; - n , β , γ , δ ; z )
is a polynomial of degree n , and hence a solution of (31.2.1) that is analytic at all three finite singularities 0 , 1 , a . These solutions are the Heun polynomials. …
3: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
Normalization
Orthogonal Invariance
Summation
Mean-Value
4: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
Euler Numbers and Polynomials
The notations E n , E n ( x ) , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
5: 18.3 Definitions
§18.3 Definitions
Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … For exact values of the coefficients of the Jacobi polynomials P n ( α , β ) ( x ) , the ultraspherical polynomials C n ( λ ) ( x ) , the Chebyshev polynomials T n ( x ) and U n ( x ) , the Legendre polynomials P n ( x ) , the Laguerre polynomials L n ( x ) , and the Hermite polynomials H n ( x ) , see Abramowitz and Stegun (1964, pp. 793–801). … For another version of the discrete orthogonality property of the polynomials T n ( x ) see (3.11.9). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
6: 17.17 Physical Applications
See Kassel (1995). … It involves q -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …
7: Bibliography G
  • W. Gautschi (1974) A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.
  • A. Gil and J. Segura (1998) A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.
  • A. Gil and J. Segura (2000) Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.
  • A. Gil and J. Segura (2003) Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.
  • S. G. Gindikin (1964) Analysis in homogeneous domains. Uspehi Mat. Nauk 19 (4 (118)), pp. 3–92 (Russian).
  • 8: 18.38 Mathematical Applications
    Approximation Theory
    Integrable Systems
    Zonal Spherical Harmonics
    Ultraspherical polynomials are zonal spherical harmonics. …
    Group Representations
    9: 12.17 Physical Applications
    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).
    10: Donald St. P. Richards
    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. …