Neumann%20polynomial
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21: 32.8 Rational Solutions
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►where the are monic polynomials (coefficient of highest power of is ) satisfying
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►Next, let be the polynomials defined by for , and
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►where and are polynomials of degree , with no common zeros.
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►where and are polynomials of degrees and , respectively, with no common zeros.
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►where , are constants, and , are polynomials of degrees and , respectively, with no common zeros.
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22: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
23: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively -binomial sums and basic hypergeometric series.
Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
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24: 24.3 Graphs
25: 18.4 Graphics
26: 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
…27: 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). …28: 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
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18.6.4
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29: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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On the real roots of Euler polynomials.
Monatsh. Math. 106 (2), pp. 115–138.
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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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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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30: 8 Incomplete Gamma and Related
Functions
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