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24 Bernoulli and Euler PolynomialsProperties

§24.13 Integrals

Contents

§24.13(i) Bernoulli Polynomials

24.13.1 Bn(t)t =Bn+1(t)n+1+const.,
24.13.2 xx+1Bn(t)t =xn,
n=1,2,,
24.13.3 xx+(1/2)Bn(t)t =En(2x)2n+1,
24.13.4 01/2Bn(t)t =1-2n+12nBn+1n+1,
24.13.5 1/43/4Bn(t)t =En22n+1.

For m,n=1,2,,

24.13.6 01Bn(t)Bm(t)t=(-1)n-1m!n!(m+n)!Bm+n.

For integrals of the form 0xBn(t)Bm(t)t and 0xBn(t)Bm(t)Bk(t)t see Agoh and Dilcher (2011).

§24.13(ii) Euler Polynomials

24.13.7 En(t)t=En+1(t)n+1+const.,
24.13.8 01En(t)t=-2En+1(0)n+1=4(2n+2-1)(n+1)(n+2)Bn+2,
24.13.9 01/2E2n(t)t=-E2n+1(0)2n+1=2(22n+2-1)B2n+2(2n+1)(2n+2),
24.13.10 01/2E2n-1(t)t=E2nn22n+1,
n=1,2,.

For m,n=1,2,,

24.13.11 01En(t)Em(t)t=(-1)n4(2m+n+2-1)m!n!(m+n+2)!Bm+n+2.

§24.13(iii) Compendia

For Laplace and inverse Laplace transforms see Prudnikov et al. (1992a, §§3.28.1–3.28.2) and Prudnikov et al. (1992b, §§3.26.1–3.26.2). For other integrals see Prudnikov et al. (1990, pp. 55–57).