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24 Bernoulli and Euler PolynomialsProperties

§24.7 Integral Representations

Contents

§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+14n1-21-2n0t2n-1e2πt+1dt
=(-1)n+12n1-21-2n0t2n-1e-πtsech(πt)dt,
24.7.2 B2n =(-1)n+14n0t2n-1e2πt-1dt
=(-1)n+12n0t2n-1e-πtcsch(πt)dt,
24.7.3 B2n =(-1)n+1π1-21-2n0t2nsech2(πt)dt,
24.7.4 B2n =(-1)n+1π0t2ncsch2(πt)dt,
24.7.5 B2n =(-1)n2n(2n-1)π0t2n-2ln(1-e-2πt)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)-e-2πtcosh(2πt)-cos(2πx)t2n-1dt,
n=1,2,,
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)=12πi-c-i-c+i(x+t)n(πsin(πt))2dt,
0<c<1.

§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).