homogeneous harmonic polynomials
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41—50 of 282 matching pages
41: 23.10 Addition Theorems and Other Identities
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§23.10(iv) Homogeneity
…42: 31.14 General Fuchsian Equation
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►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions).
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43: 18.4 Graphics
44: 19.18 Derivatives and Differential Equations
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►and two similar equations obtained by permuting in (19.18.10).
►More concisely, if , then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation:
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45: 18.7 Interrelations and Limit Relations
§18.7 Interrelations and Limit Relations
… ►Chebyshev, Ultraspherical, and Jacobi
… ►Legendre, Ultraspherical, and Jacobi
… ►§18.7(ii) Quadratic Transformations
… ►§18.7(iii) Limit Relations
…46: 18.41 Tables
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§18.41(i) Polynomials
►For () see §14.33. ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates , , , and for . The ranges of are for and , and for and . … ►For , , and see §3.5(v). …47: 18.6 Symmetry, Special Values, and Limits to Monomials
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►For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1.
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Laguerre
… ► ►§18.6(ii) Limits to Monomials
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18.6.4
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48: 18.1 Notation
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Classical OP’s
… ►Hahn Class OP’s
… ►Wilson Class OP’s
… ►Nor do we consider the shifted Jacobi polynomials: …or the dilated Chebyshev polynomials of the first and second kinds: …49: Bibliography D
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The constrained quantum mechanical harmonic oscillator.
Proc. Cambridge Philos. Soc. 62, pp. 277–286.
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On the real roots of Euler polynomials.
Monatsh. Math. 106 (2), pp. 115–138.
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On multiple zeros of Bernoulli polynomials.
Acta Arith. 134 (2), pp. 149–155.
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Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets.
SIAM J. Math. Anal. 30 (5), pp. 1029–1056.
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Uniform asymptotic expansions for Charlier polynomials.
J. Approx. Theory 112 (1), pp. 93–133.
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50: Bibliography L
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The real zeros of the Bernoulli polynomials.
J. Approx. Theory 58 (2), pp. 124–150.
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On the maxima and minima of Bernoulli polynomials.
Amer. Math. Monthly 47 (8), pp. 533–538.
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Computations of spheroidal harmonics with complex arguments: A review with an algorithm.
Phys. Rev. E 58 (5), pp. 6792–6806.
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Approximation of orthogonal polynomials in terms of Hermite polynomials.
Methods Appl. Anal. 6 (2), pp. 131–146.
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Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel, and Buchholz polynomials.
J. Math. Anal. Appl. 239 (2), pp. 457–477.
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