q-Bernoulli%20polynomials
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11: 20 Theta Functions
Chapter 20 Theta Functions
…12: 18.5 Explicit Representations
13: 24.2 Definitions and Generating Functions
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§24.2(i) Bernoulli Numbers and Polynomials
… ►§24.2(ii) Euler Numbers and Polynomials
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14: 1.11 Zeros of Polynomials
§1.11 Zeros of Polynomials
… ►Horner’s Scheme
… ►§1.11(ii) Elementary Properties
… ►The discriminant of is defined by … ►§1.11(v) Stable Polynomials
…15: 3.8 Nonlinear Equations
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§3.8(iv) Zeros of Polynomials
►The polynomial … ►Example. Wilkinson’s Polynomial
… ►Consider and . We have and . …16: Bibliography I
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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On polynomials orthogonal with respect to certain Sobolev inner products.
J. Approx. Theory 65 (2), pp. 151–175.
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Two families of associated Wilson polynomials.
Canad. J. Math. 42 (4), pp. 659–695.
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Two families of orthogonal polynomials related to Jacobi polynomials.
Rocky Mountain J. Math. 21 (1), pp. 359–375.
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Classical and Quantum Orthogonal Polynomials in One Variable.
Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
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17: 6.20 Approximations
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Cody and Thacher (1968) provides minimax rational approximations for , with accuracies up to 20S.
Cody and Thacher (1969) provides minimax rational approximations for , with accuracies up to 20S.
MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions and , with accuracies up to 20S.
18: 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)).19: 7.24 Approximations
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Hastings (1955) gives several minimax polynomial and rational approximations for , and the auxiliary functions and .
Cody (1969) provides minimax rational approximations for and . The maximum relative precision is about 20S.
Cody et al. (1970) gives minimax rational approximations to Dawson’s integral (maximum relative precision 20S–22S).
20: 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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