Digital Library of Mathematical Functions
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24 Bernoulli and Euler PolynomialsProperties

§24.5 Recurrence Relations

Contents

§24.5(i) Basic Relations

24.5.1 k=0n-1(nk)Bk(x)=nxn-1,
n=2,3,,
24.5.2 k=0n(nk)Ek(x)+En(x)=2xn,
n=1,2,.
24.5.3 k=0n-1(nk)Bk=0,
n=2,3,,
24.5.4 k=0n(2n2k)E2k=0,
n=1,2,,
24.5.5 k=0n(nk)2kEn-k+En=2.

§24.5(ii) Other Identities

24.5.6 k=2n(nk-2)Bkk=1(n+1)(n+2)-Bn+1,
n=2,3,,
24.5.7 k=0n(nk)Bkn+2-k=Bn+1n+1,
n=1,2,,
24.5.8 k=0n22kB2k(2k)!(2n+1-2k)!=1(2n)!,
n=1,2,.

§24.5(iii) Inversion Formulas

In each of (24.5.9) and (24.5.10) the first identity implies the second one and vice-versa.

24.5.9 an =k=0n(nk)bn-kk+1,
bn =k=0n(nk)Bkan-k.
24.5.10 an =k=0n/2(n2k)bn-2k,
bn =k=0n/2(n2k)E2kan-2k.