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24 Bernoulli and Euler PolynomialsProperties

§24.7 Integral Representations

  1. §24.7(i) Bernoulli and Euler Numbers
  2. §24.7(ii) Bernoulli and Euler Polynomials
  3. §24.7(iii) Compendia

§24.7(i) Bernoulli and Euler Numbers

The identities in this subsection hold for n=1,2,. (24.7.6) also holds for n=0.

24.7.1 B2n =(1)n+14n1212n0t2n1e2πt+1dt=(1)n+12n1212n0t2n1eπtsech(πt)dt,
24.7.2 B2n =(1)n+14n0t2n1e2πt1dt=(1)n+12n0t2n1eπtcsch(πt)dt,
24.7.3 B2n =(1)n+1π1212n0t2nsech2(πt)dt,
24.7.4 B2n =(1)n+1π0t2ncsch2(πt)dt,
24.7.5 B2n =(1)n2n(2n1)π0t2n2ln(1e2πt)dt.
24.7.6 E2n =(1)n22n+10t2nsech(πt)dt.

§24.7(ii) Bernoulli and Euler Polynomials

The following four equations hold for 0<x<1.

24.7.7 B2n(x) =(1)n+12n0cos(2πx)e2πtcosh(2πt)cos(2πx)t2n1dt,
24.7.8 B2n+1(x) =(1)n+1(2n+1)0sin(2πx)cosh(2πt)cos(2πx)t2ndt.
24.7.9 E2n(x) =(1)n40sin(πx)cosh(πt)cosh(2πt)cos(2πx)t2ndt,
24.7.10 E2n+1(x) =(1)n+140cos(πx)sinh(πt)cosh(2πt)cos(2πx)t2n+1dt.

Mellin–Barnes Integral

24.7.11 Bn(x)=12πicic+i(x+t)n(πsin(πt))2dt,

§24.7(iii) Compendia

For further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2000, Chapters 3 and 4).