Bernoulli polynomials
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21: 24.11 Asymptotic Approximations
22: 24.19 Methods of Computation
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§24.19(i) Bernoulli and Euler Numbers and Polynomials
… ►For algorithms for computing , , , and see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). …23: 24.6 Explicit Formulas
24: 24.5 Recurrence Relations
25: 2.10 Sums and Sequences
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►As in §24.2, let and denote the th Bernoulli number and polynomial, respectively, and the th Bernoulli periodic function .
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2.10.1
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2.10.5
►From §24.12(i), (24.2.2), and (24.4.27), is of constant sign .
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26: 24 Bernoulli and Euler Polynomials
Chapter 24 Bernoulli and Euler Polynomials
…27: 25.6 Integer Arguments
28: Errata
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Equations (25.11.6), (25.11.19), and (25.11.20)
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Equation (5.17.5)
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Originally all six integrands in these equations were incorrect because their numerators contained the function . The correct function is . The new equations are:
25.11.6
, ,
Reported 2016-05-08 by Clemens Heuberger.
25.11.19
, ,
Reported 2016-06-27 by Gergő Nemes.
25.11.20
, ,
Reported 2016-06-27 by Gergő Nemes.
5.17.5
Originally the term was incorrectly stated as .
Reported 2013-08-01 by Gergő Nemes and subsequently by Nick Jones on December 11, 2013.
29: Software Index
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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24 Bernoulli and Euler Polynomials | |||||||||||||||||||||||||
24.21(ii) , , , | ✓ | ✓ | ✓ | ✓ | a | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Derive, MuPAD | ||||||||||||
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30: Bibliography T
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Bernoulli polynomials old and new: Generalizations and asymptotics.
CWI Quarterly 8 (1), pp. 47–66.
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Explicit formulas for the Bernoulli and Euler polynomials and numbers.
Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
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On the theory of the Bernoulli polynomials and numbers.
J. Math. Anal. Appl. 104 (2), pp. 309–350.
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