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24 Bernoulli and Euler PolynomialsProperties

§24.6 Explicit Formulas

The identities in this section hold for n=1,2,. (24.6.7), (24.6.8), (24.6.10), and (24.6.12) are valid also for n=0.

24.6.1 B2n=k=22n+1(1)k1k(2n+1k)j=1k1j2n,
24.6.2 Bn=1n+1k=1nj=1k(1)jjn(n+1kj)/(nk),
24.6.3 B2n=k=1n(k1)!k!(2k+1)!j=1k(1)j1(2kk+j)j2n.
24.6.4 E2n=k=1n12k1j=1k(1)j(2kkj)j2n,
24.6.5 E2n=12n1k=0n1(1)nk(nk)2nj=0k(2n2jkj)2j,
24.6.6 E2n=k=12n(1)k2k1(2n+1k+1)j=012k12(kj)(k2j)2n.
24.6.7 Bn(x)=k=0n1k+1j=0k(1)j(kj)(x+j)n,
24.6.8 En(x)=12nk=1n+1j=0k1(1)j(n+1k)(x+j)n.
24.6.9 Bn =k=0n1k+1j=0k(1)j(kj)jn,
24.6.10 En =12nk=1n+1(n+1k)j=0k1(1)j(2j+1)n.
24.6.11 Bn=n2n(2n1)k=1nj=0k1(1)j+1(nk)jn1,
24.6.12 E2n=k=02n12kj=0k(1)j(kj)(1+2j)2n.