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1: 25.16 Mathematical Applications
§25.16(ii) Euler Sums
Euler sums have the form … For integer s ( 2 ), H ( s ) can be evaluated in terms of the zeta function: … H ( s ) is the special case H ( s , 1 ) of the function …when both H ( s , z ) and H ( z , s ) are finite. …
2: 24.14 Sums
§24.14 Sums
24.14.3 k = 0 n ( n k ) E k ( h ) E n - k ( x ) = 2 ( E n + 1 ( x + h ) - ( x + h - 1 ) E n ( x + h ) ) ,
24.14.5 k = 0 n ( n k ) E k ( h ) B n - k ( x ) = 2 n B n ( 1 2 ( x + h ) ) ,
24.14.9 ( 2 n ) ! ( 2 j ) ! ( 2 k ) ! ( 2 ) ! E 2 j E 2 k E 2 = 1 2 ( E 2 n - E 2 n + 2 ) .
For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).
3: 17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions
Euler’s Second Sum
Euler’s First Sum
4: 24.20 Tables
§24.20 Tables
5: 5.7 Series Expansions
5.7.1 1 Γ ( z ) = k = 1 c k z k ,
5.7.3 ln Γ ( 1 + z ) = - ln ( 1 + z ) + z ( 1 - γ ) + k = 2 ( - 1 ) k ( ζ ( k ) - 1 ) z k k , | z | < 2 .
5.7.4 ψ ( 1 + z ) = - γ + k = 2 ( - 1 ) k ζ ( k ) z k - 1 , | z | < 1 ,
5.7.5 ψ ( 1 + z ) = 1 2 z - π 2 cot ( π z ) + 1 z 2 - 1 + 1 - γ - k = 1 ( ζ ( 2 k + 1 ) - 1 ) z 2 k , | z | < 2 , z 0 , ± 1 .
5.7.6 ψ ( z ) = - γ - 1 z + k = 1 z k ( k + z ) = - γ + k = 0 ( 1 k + 1 - 1 k + z ) ,
6: Bibliography F
  • P. Flajolet and B. Salvy (1998) Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.
  • 7: 8.7 Series Expansions
    8.7.1 γ * ( a , z ) = e - z k = 0 z k Γ ( a + k + 1 ) = 1 Γ ( a ) k = 0 ( - z ) k k ! ( a + k ) .
    8.7.2 γ ( a , x + y ) - γ ( a , x ) = Γ ( a , x ) - Γ ( a , x + y ) = e - x x a - 1 n = 0 ( 1 - a ) n ( - x ) n ( 1 - e - y e n ( y ) ) , | y | < | x | .
    8.7.3 Γ ( a , z ) = Γ ( a ) - k = 0 ( - 1 ) k z a + k k ! ( a + k ) = Γ ( a ) ( 1 - z a e - z k = 0 z k Γ ( a + k + 1 ) ) , a 0 , - 1 , - 2 , .
    8.7.4 γ ( a , x ) = Γ ( a ) x 1 2 a e - x n = 0 e n ( - 1 ) x 1 2 n I n + a ( 2 x 1 / 2 ) , a 0 , - 1 , - 2 , .
    8.7.6 Γ ( a , x ) = x a e - x n = 0 L n ( a ) ( x ) n + 1 , x > 0 .
    8: 27.5 Inversion Formulas
    27.5.4 n = d | n ϕ ( d ) ϕ ( n ) = d | n d μ ( n d ) ,
    9: Bibliography B
  • A. Basu and T. M. Apostol (2000) A new method for investigating Euler sums. Ramanujan J. 4 (4), pp. 397–419.
  • 10: 6.15 Sums
    6.15.3 n = 1 ( - 1 ) n Ci ( 2 π n ) = 1 - ln 2 - 1 2 γ ,