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1: 24.2 Definitions and Generating Functions
§24.2(iii) Periodic Bernoulli and Euler Functions
2: 25.1 Special Notation
k , m , n

nonnegative integers.

B ~ n ( x )

periodic Bernoulli function B n ( x - x ) .

3: 25.11 Hurwitz Zeta Function
25.11.6 ζ ( s , a ) = 1 a s ( 1 2 + a s - 1 ) - s ( s + 1 ) 2 0 B ~ 2 ( x ) - B 2 ( x + a ) s + 2 d x , s 1 , s > - 1 , a > 0 .
25.11.7 ζ ( s , a ) = 1 a s + 1 ( 1 + a ) s ( 1 2 + 1 + a s - 1 ) + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k 1 ( 1 + a ) s + 2 k - 1 - ( s + 2 n 2 n + 1 ) 1 B ~ 2 n + 1 ( x ) ( x + a ) s + 2 n + 1 d x , s 1 , a > 0 , n = 1 , 2 , 3 , , s > - 2 n .
25.11.19 ζ ( s , a ) = - ln a a s ( 1 2 + a s - 1 ) - a 1 - s ( s - 1 ) 2 + s ( s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ln ( x + a ) ( x + a ) s + 2 d x - ( 2 s + 1 ) 2 0 B ~ 2 ( x ) - B 2 ( x + a ) s + 2 d x , s > - 1 , s 1 , a > 0 .
25.11.20 ( - 1 ) k ζ ( k ) ( s , a ) = ( ln a ) k a s ( 1 2 + a s - 1 ) + k ! a 1 - s r = 0 k - 1 ( ln a ) r r ! ( s - 1 ) k - r + 1 - s ( s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k ( x + a ) s + 2 d x + k ( 2 s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k - 1 ( x + a ) s + 2 d x - k ( k - 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k - 2 ( x + a ) s + 2 d x , s > - 1 , s 1 , a > 0 .
4: 24.17 Mathematical Applications
24.17.5 M n ( x ) = { B ~ n ( x ) - B n , n  even , B ~ n ( x + 1 2 ) , n  odd .
24.17.8 F ( x ) = B ~ n ( x ) - 2 - n B n
5: 25.16 Mathematical Applications
25.16.6 H ( s ) = - ζ ( s ) + γ ζ ( s ) + 1 2 ζ ( s + 1 ) + r = 1 k ζ ( 1 - 2 r ) ζ ( s + 2 r ) + n = 1 1 n s n B ~ 2 k + 1 ( x ) x 2 k + 2 d x ,
25.16.7 H ( s ) = 1 2 ζ ( s + 1 ) + ζ ( s ) s - 1 - r = 1 k ( s + 2 r - 2 2 r - 1 ) ζ ( 1 - 2 r ) ζ ( s + 2 r ) - ( s + 2 k 2 k + 1 ) n = 1 1 n n B ~ 2 k + 1 ( x ) x s + 2 k + 1 d x .
6: 25.2 Definition and Expansions
25.2.9 ζ ( s ) = k = 1 N 1 k s + N 1 - s s - 1 - 1 2 N - s + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k N 1 - s - 2 k - ( s + 2 n 2 n + 1 ) N B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > - 2 n ; n , N = 1 , 2 , 3 , .
25.2.10 ζ ( s ) = 1 s - 1 + 1 2 + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k - ( s + 2 n 2 n + 1 ) 1 B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > - 2 n , n = 1 , 2 , 3 , .
7: 2.10 Sums and Sequences
As in §24.2, let B n and B n ( x ) denote the n th Bernoulli number and polynomial, respectively, and B ~ n ( x ) the n th Bernoulli periodic function B n ( x - x ) . …
2.10.1 j = a n f ( j ) = a n f ( x ) d x + 1 2 f ( a ) + 1 2 f ( n ) + s = 1 m - 1 B 2 s ( 2 s ) ! ( f ( 2 s - 1 ) ( n ) - f ( 2 s - 1 ) ( a ) ) + a n B 2 m - B ~ 2 m ( x ) ( 2 m ) ! f ( 2 m ) ( x ) d x .
2.10.5 R m ( n ) = n B ~ 2 m ( x ) - B 2 m 2 m ( 2 m - 1 ) x 2 m - 1 d x .
8: Errata
  • Equations (25.11.6), (25.11.19), and (25.11.20)

    Originally all six integrands in these equations were incorrect because their numerators contained the function B ~ 2 ( x ) . The correct function is B ~ 2 ( x ) - B 2 2 . The new equations are:

    25.11.6 ζ ( s , a ) = 1 a s ( 1 2 + a s - 1 ) - s ( s + 1 ) 2 0 B ~ 2 ( x ) - B 2 ( x + a ) s + 2 d x , s 1 , s > - 1 , a > 0

    Reported 2016-05-08 by Clemens Heuberger.

    25.11.19 ζ ( s , a ) = - ln a a s ( 1 2 + a s - 1 ) - a 1 - s ( s - 1 ) 2 + s ( s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ln ( x + a ) ( x + a ) s + 2 d x - ( 2 s + 1 ) 2 0 B ~ 2 ( x ) - B 2 ( x + a ) s + 2 d x , s > - 1 , s 1 , a > 0

    Reported 2016-06-27 by Gergő Nemes.

    25.11.20 ( - 1 ) k ζ ( k ) ( s , a ) = ( ln a ) k a s ( 1 2 + a s - 1 ) + k ! a 1 - s r = 0 k - 1 ( ln a ) r r ! ( s - 1 ) k - r + 1 - s ( s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k ( x + a ) s + 2 d x + k ( 2 s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k - 1 ( x + a ) s + 2 d x - k ( k - 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k - 2 ( x + a ) s + 2 d x , s > - 1 , s 1 , a > 0

    Reported 2016-06-27 by Gergő Nemes.

  • 9: 24.16 Generalizations
    In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); p -adic integer order Bernoulli numbers (Adelberg (1996)); p -adic q -Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).
    10: Bibliography B
  • B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.