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Bernoulli numbers

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1: 24.1 Special Notation
Bernoulli Numbers and Polynomials
The origin of the notation B n , B n ( x ) , is not clear. …
2: 24.14 Sums
§24.14 Sums
24.14.2 k = 0 n ( n k ) B k B n - k = ( 1 - n ) B n - n B n - 1 .
24.14.6 k = 0 n ( n k ) 2 k B k E n - k = 2 ( 1 - 2 n - 1 ) B n - n E n - 1 .
24.14.8 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! B 2 j B 2 k B 2 = ( n - 1 ) ( 2 n - 1 ) B 2 n + n ( n - 1 2 ) B 2 n - 2 ,
24.14.10 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! ( 2 m ) ! B 2 j B 2 k B 2 B 2 m = - ( 2 n + 3 3 ) B 2 n - 4 3 n 2 ( 2 n - 1 ) B 2 n - 2 .
3: 24.19 Methods of Computation
Equations (24.5.3) and (24.5.4) enable B n and E n to be computed by recurrence. …
B 2 n = N 2 n D 2 n .
For algorithms for computing B n , E n , B n ( x ) , and E n ( x ) see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).
§24.19(ii) Values of B n Modulo p
For number-theoretic applications it is important to compute B 2 n ( mod p ) for 2 n p - 3 ; in particular to find the irregular pairs ( 2 n , p ) for which B 2 n 0 ( mod p ) . …
4: 24.10 Arithmetic Properties
24.10.1 B 2 n + ( p - 1 ) | 2 n 1 p = integer ,
The denominator of B 2 n is the product of all these primes p .
24.10.2 p B 2 n p - 1 ( mod p + 1 ) ,
24.10.3 B m m B n n ( mod p ) ,
§24.10(iii) Voronoi’s Congruence
5: 24.5 Recurrence Relations
§24.5(ii) Other Identities
24.5.6 k = 2 n ( n k - 2 ) B k k = 1 ( n + 1 ) ( n + 2 ) - B n + 1 , n = 2 , 3 , ,
24.5.7 k = 0 n ( n k ) B k n + 2 - k = B n + 1 n + 1 , n = 1 , 2 , ,
24.5.8 k = 0 n 2 2 k B 2 k ( 2 k ) ! ( 2 n + 1 - 2 k ) ! = 1 ( 2 n ) ! , n = 1 , 2 , .
§24.5(iii) Inversion Formulas
6: 4.19 Maclaurin Series and Laurent Series
In (4.19.3)–(4.19.9), B n are the Bernoulli numbers and E n are the Euler numbers (§§24.2(i)24.2(ii)).
4.19.3 tan z = z + z 3 3 + 2 15 z 5 + 17 315 z 7 + + ( - 1 ) n - 1 2 2 n ( 2 2 n - 1 ) B 2 n ( 2 n ) ! z 2 n - 1 + , | z | < 1 2 π ,
4.19.4 csc z = 1 z + z 6 + 7 360 z 3 + 31 15120 z 5 + + ( - 1 ) n - 1 2 ( 2 2 n - 1 - 1 ) B 2 n ( 2 n ) ! z 2 n - 1 + , 0 < | z | < π ,
4.19.6 cot z = 1 z - z 3 - z 3 45 - 2 945 z 5 - - ( - 1 ) n - 1 2 2 n B 2 n ( 2 n ) ! z 2 n - 1 - , 0 < | z | < π ,
4.19.7 ln ( sin z z ) = n = 1 ( - 1 ) n 2 2 n - 1 B 2 n n ( 2 n ) ! z 2 n , | z | < π ,
7: 24.21 Software
§24.21(ii) B n , B n ( x ) , E n , and E n ( x )
8: 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
9: 4.33 Maclaurin Series and Laurent Series
4.33.3 tanh z = z - z 3 3 + 2 15 z 5 - 17 315 z 7 + + 2 2 n ( 2 2 n - 1 ) B 2 n ( 2 n ) ! z 2 n - 1 + , | z | < 1 2 π .
For B 2 n see §24.2(i). …
10: 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 ) ,
24.6.9 B n = k = 0 n 1 k + 1 j = 0 k ( - 1 ) j ( k j ) j n ,
24.6.11 B n = n 2 n ( 2 n - 1 ) k = 1 n j = 0 k - 1 ( - 1 ) j + 1 ( n k ) j n - 1 ,