homogeneous harmonic polynomials

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31: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

32: 9.16 Physical Applications

… ►An example from quantum mechanics is given in Landau and Lifshitz (1965), in which the exact solution of the Schrödinger equation for the motion of a particle in a homogeneous external field is expressed in terms of $\mathrm{Ai}\left(x\right)$. …

33: Bibliography W

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Global asymptotics of the Meixner polynomials. Asymptotic Analysis 75 (3-4), pp. 211–231.

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External Links:

ISSN 0921-7134, Document, MathReview (Mario Pérez Riera), ZentralBlatt

Referenced by:

§18.24, §18.24.

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Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Li Zhong Peng), ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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Harmonic Analysis. In Studies in Real and Complex Analysis, I. I. Hirschman (Ed.), Studies in Mathematics, Vol. 3, pp. 124–178.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.15(iii), §1.15(iv), §1.15(v).

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On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.

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External Links:

Document, ZentralBlatt

Referenced by:

§12.1, §12.5(i), §12.5(ii), §12.9(i).

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

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34: 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. …

35: Bibliography

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SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.

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Notes:

SPHEREPACK 2.0 is a collection of Fortran programs that facilitates computer modeling of geophysical processes. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms. The package also contains utility programs for computing the associated Legendre functions. SPHEREPACK 3.1 was released in August 2003.

External Links:

FullText, SPHEREPACK 3.1

Referenced by:

1st item.

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Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.

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External Links:

ISSN 0025-584X, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.27(vii).

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Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.

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External Links:

ISSN 0002-9947, FullText, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§31.15(ii).

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A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.

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Notes:

For corrigendum see same journal v. 90 (1998), no. 2, p. 265

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54 (319), pp. iv+55.

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External Links:

ISSN 0065-9266, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.21(ii), §18.22(ii), (18.28.24), §18.28(ii), Ch.18, Profile Richard A. Askey.

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36: Bibliography H

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Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

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External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.16.

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Orthogonal Laurent polynomials. Nederl. Akad. Wetensch. Indag. Math. 48 (1), pp. 17–36.

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External Links:

ISSN 0019-3577, Document, MathReview (W. J. Thron), ZentralBlatt

Referenced by:

§18.33(v).

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The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.

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Notes:

Reprinted by Chelsea Publishing Company, New York, 1955 and 1965.

External Links:

ISBN 0-8284-0104-7, MathReview, ZentralBlatt

Referenced by:

§14.1, §14.11, §14.16(ii), §14.16(iii), §14.20(x), §14.27, §14.27, §14.3(i), §14.3(i), Ch.14, §29.18(iii), Ch.29.

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Roots of the Euler polynomials. Pacific J. Math. 64 (1), pp. 181–191.

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External Links:

MathReview (M. Marden), ZentralBlatt

Referenced by:

§24.12(ii).

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Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

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Notes:

Reprinted in 1979.

External Links:

ISBN 0-8218-1556-3, MathReview, ZentralBlatt

Referenced by:

§35.4(i).

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37: 24.3 Graphs

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Figure 24.3.1: Bernoulli polynomials $B_n\left(x\right)$, $n=2,3,\mathrm{\dots },6$. Magnify

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Figure 24.3.2: Euler polynomials $E_n\left(x\right)$, $n=2,3,\mathrm{\dots },6$. Magnify

38: 15.11 Riemann’s Differential Equation

… ►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … ►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …

39: 19.33 Triaxial Ellipsoids

… ►Let a homogeneous magnetic ellipsoid with semiaxes $a,b,c$, volume $V=4\pi abc/3$, and susceptibility $\chi$ be placed in a previously uniform magnetic field $H$ parallel to the principal axis with semiaxis $c$. … ►The same result holds for a homogeneous dielectric ellipsoid in an electric field. …

40: 20.9 Relations to Other Functions

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