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1: 24.2 Definitions and Generating Functions
§24.2(iii) Periodic Bernoulli and Euler Functions
2: 24.17 Mathematical Applications
24.17.3 S n ( x ) = E ~ n ( x + 1 2 n + 1 2 ) E ~ n ( 1 2 n + 1 2 ) , n = 0 , 1 , ,
3: 25.13 Periodic Zeta Function
25.13.2 F ( x , s ) = Γ ( 1 - s ) ( 2 π ) 1 - s ( e π i ( 1 - s ) / 2 ζ ( 1 - s , x ) + e π i ( s - 1 ) / 2 ζ ( 1 - s , 1 - x ) ) , 0 < x < 1 , s > 1 ,
25.13.3 ζ ( 1 - s , x ) = Γ ( s ) ( 2 π ) s ( e - π i s / 2 F ( x , s ) + e π i s / 2 F ( - x , s ) ) , s > 0 if 0 < x < 1 ; s > 1 if x = 1 .
4: 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 .
5: 25.1 Special Notation
(For other notation see Notation for the Special Functions.)
k , m , n

nonnegative integers.

B ~ n ( x )

periodic Bernoulli function B n ( x - x ) .

The main function treated in this chapter is the Riemann zeta function ζ ( s ) . … The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
6: 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)).
7: 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 .
8: 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 , .
9: 32.10 Special Function Solutions
§32.10 Special Function Solutions
If γ δ 0 , then as in §32.2(ii) we may set γ = 1 and δ = - 1 . … where the fundamental periods 2 ϕ 1 and 2 ϕ 2 are linearly independent functions satisfying the hypergeometric equation …The solution (32.10.34) is an essentially transcendental function of both constants of integration since P VI  with α = β = γ = 0 and δ = 1 2 does not admit an algebraic first integral of the form P ( z , w , w , C ) = 0 , with C a constant. …
10: 2.10 Sums and Sequences
§2.10(i) Euler–Maclaurin Formula
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 ) . … From §24.12(i), (24.2.2), and (24.4.27), B ~ 2 m ( x ) - B 2 m is of constant sign ( - 1 ) m . … where γ is Euler’s constant (§5.2(ii)) and ζ is the derivative of the Riemann zeta function25.2(i)). … For extensions of the Euler–Maclaurin formula to functions f ( x ) with singularities at x = a or x = n (or both) see Sidi (2004, 2012b, 2012a). …