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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

… ► $Y_l^m(\theta ,\varphi )$ are known as surface harmonics of the first kind: tesseral for and sectorial for $|m|=l$. … … ►

2: 20 Theta Functions

Chapter 20 Theta Functions

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3: Errata

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The following additions were made in Chapter 18:

• In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)–18.2(xii), with Equations (18.2.21)–(18.2.46),

• A new introduction, minor changes in the presentation, and three new paragraphs.

• Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).

• Line numbers and two extra rows were added to Table 18.8.1.

• Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.

• Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).

• Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).

• Equation (18.15.4_5).

• The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).

• Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).

• Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.

• A new introduction.

• Equation (18.21.13).

• Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.

• In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).

• Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).

• A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)–18.28(xi), with Equations (18.28.23)–(18.28.34).

• Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)–18.30(viii), with Equations (18.30.8)–(18.30.31).

• This short section has been expanded, with Equation (18.32.2).

• Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).

• This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).

• This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).

• This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.

• The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).

• This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

  The original text of 18.39(i) Quantum Mechanics was:

  “Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$:

  errata.1 $$\left(\frac{-{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}+V(x)\right)\psi (x,t)=i\mathrm{\hslash }\frac{\partial }{\partial t}\psi (x,t),$$

  where $\mathrm{\hslash }$ is the reduced Planck’s constant. On substituting $\psi (x,t)=\eta (x)\zeta (t)$, we obtain two ordinary differential equations, each of which involve the same constant $E$. The equation for $\eta (x)$ is

  errata.2 $$\frac{d^2\eta }{{dx}^2}+\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(x)\right)\eta =0.$$

  For a harmonic oscillator, the potential energy is given by

  errata.3 $$V(x)=\frac{1}{2}m{\omega }^2{x}^2,$$

  where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval , the constant $E$ (the total energy of the particle) must satisfy

  errata.4 $$E=E_n=\left(n+\frac{1}{2}\right)\mathrm{\hslash }\omega ,$$ $n=0,1,2,\mathrm{\dots }$.

  The corresponding eigenfunctions are

  errata.5 $${\eta }_n(x)={\pi }^{-\frac{1}{4}}{2}^{-\frac{1}{2}n}{(n!b)}^{-\frac{1}{2}}{H}_n\left(x/b\right){\mathrm{e}}^{-x^2/2b^2},$$

  where $b={(\mathrm{\hslash }/m\omega )}^{1/2}$, and $H_n$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

  A second example is provided by the three-dimensional time-independent Schrödinger equation

  errata.6 $${\nabla }^2\psi +\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(𝐱)\right)\psi =0,$$

  when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

  For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”

• The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

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Previously the notation $h(n)$ was used for the harmonic number $H_n$ (defined in (25.11.33)). The more widely used notation $H_n$ will now be used throughout the DLMF. In particular, this change was made in (25.11.32), (25.11.33), (25.16.5) and (25.16.13) (suggested by Gergő Nemes on 2021-08-23).

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In regard to the definition of the spherical harmonics $Y_{l,m}$, the domain of the integer $m$ originally written as $0\le m\le l$ has been replaced with the more general $|m|\le l$. Because of this change, in the sentence just below (14.30.2), “tesseral for and sectorial for $m=l$” has been replaced with “tesseral for and sectorial for $|m|=l$”. Furthermore, in (14.30.4), $m$ has been replaced with $|m|$.

Reported by Ching-Li Chai on 2019-10-05

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

… ►

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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4: 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. …

5: 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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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.

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Notes:

Double-precision Fortran, maximum accuracy 18S

External Links:

Document, Code, ZentralBlatt

Referenced by:

5th item, 1st item.

Encodings:

BibTeX

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Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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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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External Links:

ISSN 1751-8113, Document, Link, MathReview (Brian Simanek)

Referenced by:

§18.36(vi), §18.36(vi).

Encodings:

BibTeX

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6: 25.11 Hurwitz Zeta Function

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Figure 25.11.1: Hurwitz zeta function $\zeta (x,a)$, $a$ = 0. …8, 1, $-20\le x\le 10$. … Magnify

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25.11.32 $${\int }_0^a{x}^n\psi \left(x\right)dx={(-1)}^{n-1}{\zeta }^{\prime }\left(-n\right)+{(-1)}^n{H}_n\frac{B_{n+1}}{n+1}-\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)H_k\frac{B_{k+1}(a)}{k+1}{a}^{n-k}+\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){\zeta }^{\prime }(-k,a)a^{n-k},$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$,

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $H_n$: harmonic number, $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $h(n)$: harmonic number

Keywords:

definite integral, integral representation

Source:

Adamchik (1998, (17), p. 197)

Referenced by:

Erratum (V1.1.4) for Notation

Permalink:

http://dlmf.nist.gov/25.11.E32

Encodings:

pMML, png, TeX

Notation (effective with 1.1.4):

The notation previously used for the harmonic number $h(n)$ (resp. $h(k)$) has been replaced to be $H_n$ (resp. $H_k$).

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

►where $H_n$ are the harmonic numbers: ►

25.11.33 $$H_n=\sum _{k=1}^n{k}^{-1}.$$

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Defines:

$H_n$: harmonic number

Symbols:

$k$: nonnegative integer, $n$: nonnegative integer and $h(n)$: harmonic number

Keywords:

harmonic number

Source:

Knuth (1968, Section 1.2.7, p. 73)

Referenced by:

§25.11(viii), §25.16(ii), Erratum (V1.1.4) for Notation

Permalink:

http://dlmf.nist.gov/25.11.E33

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 §25.11(viii), §25.11 and Ch.25

…

7: 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. …

8: 18.39 Applications in the Physical Sciences

… ►argument a) The Harmonic Oscillator … ►This is illustrated in Figure 18.39.1 where the first and fourth excited state eigenfunctions of the Schrödinger operator with the rationally extended harmonic potential, of (18.39.19), are shown, and compared with the first and fourth excited states of the harmonic oscillator eigenfunctions of (18.39.14) of paragraph a), above. … ►The eigenfunctions of ${\mathrm{L}}^2$ are the spherical harmonics $Y_{l,m_l}(\theta ,\varphi )$ with eigenvalues ${\mathrm{\hslash }}^{2}l(l+1)$, each with degeneracy $2l+1$ as $m_l=-l,-l+1,\mathrm{\dots },l$. … … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …

9: 15.17 Mathematical Applications

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§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. …

10: Bibliography M

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

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New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..

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External Links:

MathReview Entry

Referenced by:

§18.36(vi).

Encodings:

BibTeX

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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External Links:

ISSN 1063-5203, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(iii).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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