homogeneous harmonic polynomials
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21: 18.39 Applications in the Physical Sciences
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► a) The Harmonic Oscillator
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►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.
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►The eigenfunctions of are the spherical harmonics
with eigenvalues , each with degeneracy as .
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The Coulomb–Pollaczek Polynomials
…22: 2.9 Difference Equations
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►or equivalently the second-order homogeneous linear difference equation
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►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)).
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►For applications of asymptotic methods for difference equations to orthogonal polynomials, see, e.
…These methods are particularly useful when the weight function associated with the orthogonal polynomials is not unique or not even known; see, e.
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23: 1.17 Integral and Series Representations of the Dirac Delta
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Legendre Polynomials (§§14.7(i) and 18.3)
… ►Laguerre Polynomials (§18.3)
… ►Hermite Polynomials (§18.3)
… ►Spherical Harmonics (§14.30)
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1.17.25
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24: 3.6 Linear Difference Equations
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►If , , then the difference equation is homogeneous; otherwise it is inhomogeneous.
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§3.6(ii) Homogeneous Equations
… ►Because the recessive solution of a homogeneous equation is the fastest growing solution in the backward direction, it occurred to J. … ►See also Gautschi (1967) and Gil et al. (2007a, Chapter 4) for the computation of recessive solutions via continued fractions. … ►It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. …25: Bibliography K
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Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials.
Appl. Anal. 90 (3-4), pp. 731–746.
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The addition formula for Laguerre polynomials.
SIAM J. Math. Anal. 8 (3), pp. 535–540.
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Meixner-Pollaczek polynomials and the Heisenberg algebra.
J. Math. Phys. 30 (4), pp. 767–769.
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Askey-Wilson polynomial.
Scholarpedia 7 (7), pp. 7761.
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An algorithm for solving second order linear homogeneous differential equations.
J. Symbolic Comput. 2 (1), pp. 3–43.
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26: 25.11 Hurwitz Zeta Function
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►For see §24.2(iii).
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25.11.14
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25.11.32
, ,
►where are the harmonic numbers:
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25.11.33
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27: Bibliography T
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Zonal Polynomials.
Institute of Mathematical Statistics Lecture Notes—Monograph
Series, 4, Institute of Mathematical Statistics, Hayward, CA.
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Laguerre polynomials: Asymptotics for large degree.
Technical report
Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.
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Harmonic Analysis on Symmetric Spaces and Applications. II.
Springer-Verlag, Berlin.
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Hyperspherical elliptic harmonics and their relation to the Heun equation.
Phys. Rev. A 63 (032510), pp. 1–8.
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Representation Theory and Harmonic Analysis.
Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.
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28: 21.7 Riemann Surfaces
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21.7.1
►where is a polynomial in and that does not factor over .
…To accomplish this we write (21.7.1) in terms of homogeneous coordinates:
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21.7.11
►where is a polynomial in of odd degree
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29: 1.2 Elementary Algebra
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►Let be distinct constants, and be a polynomial of degree less than .
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►To find the polynomials
, , multiply both sides by the denominator of the left-hand side and equate coefficients.
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