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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: 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)).
9: 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.

  • 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.