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tesseral harmonics

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1: 14.30 Spherical and Spheroidal Harmonics
§14.30 Spherical and Spheroidal Harmonics
§14.30(i) Definitions
Y l m ( θ , ϕ ) are known as surface harmonics of the first kind: tesseral for | m | < l and sectorial for | m | = l . … …
2: 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. …
3: 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. …
4: 15.17 Mathematical Applications
§15.17(iii) Group Representations
For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …
5: 18.38 Mathematical Applications
Zonal Spherical Harmonics
Ultraspherical polynomials are zonal spherical harmonics. …
6: Tom H. Koornwinder
Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …
7: 29.18 Mathematical Applications
§29.18(iii) Spherical and Ellipsoidal Harmonics
8: 34.3 Basic Properties: 3 j Symbol
§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
For the polynomials P l see §18.3, and for the function Y l , m see §14.30. …
34.3.20 Y l 1 , m 1 ( θ , ϕ ) Y l 2 , m 2 ( θ , ϕ ) = l , m ( ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l + 1 ) 4 π ) 1 2 ( l 1 l 2 l m 1 m 2 m ) Y l , m ( θ , ϕ ) ¯ ( l 1 l 2 l 0 0 0 ) ,
34.3.22 0 2 π 0 π Y l 1 , m 1 ( θ , ϕ ) Y l 2 , m 2 ( θ , ϕ ) Y l 3 , m 3 ( θ , ϕ ) sin θ d θ d ϕ = ( ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) 4 π ) 1 2 ( l 1 l 2 l 3 0 0 0 ) ( l 1 l 2 l 3 m 1 m 2 m 3 ) .
9: 1.2 Elementary Algebra
§1.2(iv) Means
The geometric mean G and harmonic mean H of n positive numbers a 1 , a 2 , , a n are given by …
1.2.19 1 H = 1 n ( 1 a 1 + 1 a 2 + + 1 a n ) .
M ( - 1 ) = H ,
10: 12.17 Physical Applications
Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. … 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).