Euler%20polynomials
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1: 25.6 Integer Arguments
2: Bibliography L
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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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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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions.
Stud. Appl. Math. 103 (3), pp. 241–258.
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Large degree asymptotics of generalized Bernoulli and Euler polynomials.
J. Math. Anal. Appl. 363 (1), pp. 197–208.
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3: Software Index
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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5.24(ii) , | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | FDLIBM |
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18 Orthogonal Polynomials | |||||||||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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24 Bernoulli and Euler Polynomials | |||||||||||||||||||||||||
24.21(ii) , , , | ✓ | ✓ | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Derive, MuPAD | ||||||||||||
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4: 24.2 Definitions and Generating Functions
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§24.2(i) Bernoulli Numbers and Polynomials
… ►§24.2(ii) Euler Numbers and Polynomials
… ►§24.2(iii) Periodic Bernoulli and Euler Functions
… ► …5: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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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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6: 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.
7: 5.11 Asymptotic Expansions
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►The scaled gamma function is defined in (5.11.3) and its main property is as in the sector .
Wrench (1968) gives exact values of up to .
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5.11.8
►where is fixed, and is the Bernoulli polynomial defined in §24.2(i).
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►In terms of generalized Bernoulli polynomials
(§24.16(i)), we have for ,
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8: 25.11 Hurwitz Zeta Function
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§25.11(iii) Representations by the Euler–Maclaurin Formula
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25.11.6
, , .
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25.11.7
, , , .
►For see §24.2(iii).
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9: 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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►In the general case assume , so that as in §32.2(ii) we may set and .
…where and are polynomials of degree , 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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10: Bibliography P
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Zonal Polynomials of Order Through
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In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.),
Vol. 2, pp. 199–388.
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Orthogonal polynomials and some -beta integrals of Ramanujan.
J. Math. Anal. Appl. 112 (2), pp. 517–540.
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A new basis for the representation of the rotation group. Lamé and Heun polynomials.
J. Mathematical Phys. 14 (8), pp. 1130–1139.
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Tables of the Incomplete -function.
Biometrika Office, Cambridge University Press, Cambridge.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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