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1: 14.30 Spherical and Spheroidal Harmonics

§14.30 Spherical and Spheroidal Harmonics

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§14.30(i) Definitions

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§14.30(iii) Sums

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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: 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. …

6: 29.18 Mathematical Applications

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§29.18(iii) Spherical and Ellipsoidal Harmonics

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7: 34.3 Basic Properties: $\mathit{3j}$ Symbol

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§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}}}\left(\theta,\phi\right)Y_{{l_{2}},{m_{2}}}\left(\theta,% \phi\right)=\sum_{l,m}\left(\frac{(2l_{1}+1)(2l_{2}+1)(2l+1)}{4\pi}\right)^{% \frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l\\ m_{1}&m_{2}&m\end{pmatrix}\overline{Y_{{l},{m}}\left(\theta,\phi\right)}\begin% {pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix},

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E20
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the spherical harmonic in the sum over $l, m$ has been explicity marked up as a complex conjugate.
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.22

\int_{0}^{2\pi}\!\int_{0}^{\pi}Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{% l_{2}},{m_{2}}}\left(\theta,\phi\right)Y_{{l_{3}},{m_{3}}}\left(\theta,\phi% \right)\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi=\left(\frac{(2l_{1}+1)(2l_% {2}+1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §14.30(ii), §34.3(vii)
Permalink:: http://dlmf.nist.gov/34.3.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

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8: 25.16 Mathematical Applications

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25.16.5

H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},

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Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $H_{\NVar{n}}$ : harmonic number, $n$ : nonnegative integer, $s$ : complex variable and $h(n)$ : harmonic number
Keywords:: Euler sum, infinite series, series representation
Source:: Apostol and Vu (1984, p. 85)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.16.E5
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

►where

H_{n}

is given by (25.11.33). … ►

25.16.13

\sum_{n=1}^{\infty}\left(\frac{H_{n}}{n}\right)^{2}=\frac{17}{4}\zeta\left(4% \right),

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $H_{\NVar{n}}$ : harmonic number, $n$ : nonnegative integer and $h(n)$ : harmonic number
Keywords:: harmonic number, infinite series, special value
Source:: Flajolet and Salvy (1998, (4-1), p. 24)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.16.E13
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

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9: 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).

10: 1.7 Inequalities

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§1.7(iii) Means

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1.7.7

H\leq G\leq A,

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Symbols:: $A$ : arithmetic mean, $G$ : geometric mean and $H$ : harmonic mean
A&S Ref:: 3.2.1
Permalink:: http://dlmf.nist.gov/1.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §1.7(iii), §1.7 and Ch.1

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