Euler numbers
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11: 17.3 -Elementary and -Special Functions
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§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers
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17.3.8
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βΊThe are always polynomials in , and the are polynomials in for .
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12: 24.19 Methods of Computation
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§24.19(i) Bernoulli and Euler Numbers and Polynomials
βΊEquations (24.5.3) and (24.5.4) enable and to be computed by recurrence. …A similar method can be used for the Euler numbers based on (4.19.5). … βΊFor algorithms for computing , , , and see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). …13: 24.4 Basic Properties
14: 24.13 Integrals
15: 4.19 Maclaurin Series and Laurent Series
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βΊIn (4.19.3)–(4.19.9), are the Bernoulli numbers and are the Euler numbers (§§24.2(i)–24.2(ii)).
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4.19.5
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16: 24.7 Integral Representations
17: 24.8 Series Expansions
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24.8.9
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18: 24.15 Related Sequences of Numbers
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24.15.12
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19: 24.17 Mathematical Applications
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§24.17(iii) Number Theory
βΊBernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); -adic analysis (Koblitz (1984, Chapter 2)). …20: 24.16 Generalizations
§24.16 Generalizations
… βΊWhen they reduce to the Bernoulli and Euler numbers of order : … βΊ
24.16.13
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