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11: 24.21 Software
§24.21(ii) B n , B n ( x ) , E n , and E n ( x )
12: 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
13: 25.1 Special Notation
k , m , n nonnegative integers.
B n , B n ( x ) Bernoulli number and polynomial (§24.2(i)).
B ~ n ( x ) periodic Bernoulli function B n ( x x ) .
14: 24.17 Mathematical Applications
§24.17 Mathematical Applications
Bernoulli Monosplines
§24.17(iii) Number Theory
Bernoulli 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 L -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)); p -adic analysis (Koblitz (1984, Chapter 2)).
15: 24.9 Inequalities
§24.9 Inequalities
24.9.1 | B 2 n | > | B 2 n ( x ) | , 1 > x > 0 ,
24.9.2 ( 2 2 1 2 n ) | B 2 n | | B 2 n ( x ) B 2 n | , 1 x 0 .
24.9.4 2 ( 2 n + 1 ) ! ( 2 π ) 2 n + 1 > ( 1 ) n + 1 B 2 n + 1 ( x ) > 0 , n = 2 , 3 , ,
24.9.6 5 π n ( n π e ) 2 n > ( 1 ) n + 1 B 2 n > 4 π n ( n π e ) 2 n ,
16: 24.6 Explicit Formulas
§24.6 Explicit Formulas
24.6.1 B 2 n = k = 2 2 n + 1 ( 1 ) k 1 k ( 2 n + 1 k ) j = 1 k 1 j 2 n ,
24.6.2 B n = 1 n + 1 k = 1 n j = 1 k ( 1 ) j j n ( n + 1 k j ) / ( n k ) ,
24.6.3 B 2 n = k = 1 n ( k 1 ) ! k ! ( 2 k + 1 ) ! j = 1 k ( 1 ) j 1 ( 2 k k + j ) j 2 n .
24.6.9 B n = k = 0 n 1 k + 1 j = 0 k ( 1 ) j ( k j ) j n ,
17: 24.7 Integral Representations
§24.7(i) Bernoulli and Euler Numbers
24.7.1 B 2 n = ( 1 ) n + 1 4 n 1 2 1 2 n 0 t 2 n 1 e 2 π t + 1 d t = ( 1 ) n + 1 2 n 1 2 1 2 n 0 t 2 n 1 e π t sech ( π t ) d t ,
24.7.3 B 2 n = ( 1 ) n + 1 π 1 2 1 2 n 0 t 2 n sech 2 ( π t ) d t ,
24.7.4 B 2 n = ( 1 ) n + 1 π 0 t 2 n csch 2 ( π t ) d t ,
§24.7(ii) Bernoulli and Euler Polynomials
18: 24.15 Related Sequences of Numbers
§24.15(i) Genocchi Numbers
24.15.2 G n = 2 ( 1 2 n ) B n .
§24.15(ii) Tangent Numbers
24.15.4 T 2 n 1 = ( 1 ) n 1 2 2 n ( 2 2 n 1 ) 2 n B 2 n , n = 1 , 2 , ,
§24.15(iii) Stirling Numbers
19: 24 Bernoulli and Euler Polynomials
Chapter 24 Bernoulli and Euler Polynomials
20: Karl Dilcher
Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …