as Bernoulli or Euler polynomials
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21: 25.1 Special Notation
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nonnegative integers. | |
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Bernoulli number and polynomial (§24.2(i)). | |
periodic Bernoulli function . | |
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22: 17.3 -Elementary and -Special Functions
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§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers
►-Bernoulli Polynomials
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17.3.7
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►The are, in fact, rational functions of , and not necessarily polynomials.
The are always polynomials in , and the are polynomials in for .
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23: 5.11 Asymptotic Expansions
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►For the Bernoulli numbers , see §24.2(i).
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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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5.11.17
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24: 25.11 Hurwitz Zeta Function
25: 25.2 Definition and Expansions
26: 25.6 Integer Arguments
27: Bibliography D
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Sur les zéros réels des polynômes de Bernoulli.
Ann. Inst. Fourier (Grenoble) 41 (2), pp. 267–309 (French).
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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials.
J. Approx. Theory 49 (4), pp. 321–330.
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Zeros of Bernoulli, generalized Bernoulli and Euler polynomials.
Mem. Amer. Math. Soc. 73 (386), pp. iv+94.
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Sums of products of Bernoulli numbers.
J. Number Theory 60 (1), pp. 23–41.
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Bernoulli Numbers and Confluent Hypergeometric Functions.
In Number Theory for the Millennium, I (Urbana, IL, 2000),
pp. 343–363.
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28: 24.10 Arithmetic Properties
§24.10 Arithmetic Properties
►§24.10(i) Von Staudt–Clausen Theorem
… ►§24.10(ii) Kummer Congruences
… ►§24.10(iii) Voronoi’s Congruence
… ►§24.10(iv) Factors
…29: 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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2.10.4
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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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