What's New
About the Project
NIST
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)k-1k(2n+1k)j=1k-1j2n,
24.6.2 Bn=1n+1k=1nj=1k(-1)jjn(n+1k-j)/(nk),
24.6.3 B2n=k=1n(k-1)!k!(2k+1)!j=1k(-1)j-1(2kk+j)j2n.
24.6.4 E2n=k=1n12k-1j=1k(-1)j(2kk-j)j2n,
24.6.5 E2n=12n-1k=0n-1(-1)n-k(n-k)2nj=0k(2n-2jk-j)2j,
24.6.6 E2n=k=12n(-1)k2k-1(2n+1k+1)j=012k-12(kj)(k-2j)2n.
24.6.7 Bn(x)=k=0n1k+1j=0k(-1)j(kj)(x+j)n,
24.6.8 En(x)=12nk=1n+1j=0k-1(-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=0k-1(-1)j(2j+1)n.
24.6.11 Bn=n2n(2n-1)k=1nj=0k-1(-1)j+1(nk)jn-1,
24.6.12 E2n=k=02n12kj=0k(-1)j(kj)(1+2j)2n.