Digital Library of Mathematical Functions
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24 Bernoulli and Euler PolynomialsProperties

§24.4 Basic Properties

Contents

§24.4(i) Difference Equations

24.4.1 Bn(x+1)-Bn(x) =nxn-1,
24.4.2 En(x+1)+En(x) =2xn.

§24.4(ii) Symmetry

24.4.3 Bn(1-x) =(-1)nBn(x),
24.4.4 En(1-x) =(-1)nEn(x).
24.4.5 (-1)nBn(-x) =Bn(x)+nxn-1,
24.4.6 (-1)n+1En(-x) =En(x)-2xn.

§24.4(iii) Sums of Powers

24.4.7 k=1mkn =Bn+1(m+1)-Bn+1n+1,
24.4.8 k=1m(-1)m-kkn =En(m+1)+(-1)mEn(0)2.
24.4.9 k=0m-1(a+dk)n =dnn+1(Bn+1(m+ad)-Bn+1(ad)),
24.4.10 k=0m-1(-1)k(a+dk)n =dn2((-1)m-1En(m+ad)+En(ad)).
24.4.11 k=1(k,m)=1mkn =1n+1j=1n+1(n+1j)(p|m(1-pn-j)Bn+1-j)mj.

§24.4(iv) Finite Expansions

24.4.12 Bn(x+h) =k=0n(nk)Bk(x)hn-k,
24.4.13 En(x+h) =k=0n(nk)Ek(x)hn-k,
24.4.14 En-1(x) =2nk=0n(nk)(1-2k)Bkxn-k,
24.4.15 B2n =2n22n(22n-1)k=0n-1(2n-12k)E2k,
24.4.16 E2n =12n+1-k=1n(2n2k-1)22k(22k-1-1)B2kk,
24.4.17 E2n =1-k=1n(2n2k-1)22k(22k-1)B2k2k.

§24.4(v) Multiplication Formulas

Raabe’s Theorem

24.4.18 Bn(mx)=mn-1k=0m-1Bn(x+km).

Next,

24.4.19 En(mx)=-2mnn+1k=0m-1(-1)kBn+1(x+km),
m=2,4,6,,
24.4.20 En(mx)=mnk=0m-1(-1)kEn(x+km),
m=1,3,5,.
24.4.21 Bn(x) =2n-1(Bn(12x)+Bn(12x+12)),
24.4.22 En-1(x) =2n(Bn(x)-2nBn(12x)),
24.4.23 En-1(x) =2nn(Bn(12x+12)-Bn(12x)),
24.4.24 Bn(mx)=mnBn(x)+nk=1nj=0k-1(-1)j(nk)(r=1m-12π(k-j)r/m(1-2πr/m)n)(j+mx)n-1,
n=1,2,, m=2,3,.

§24.4(vi) Special Values

24.4.25 Bn(0)=(-1)nBn(1)=Bn.
24.4.26 En(0)=-En(1)=-2n+1(2n+1-1)Bn+1,
n>0.
24.4.27 Bn(12) =-(1-21-n)Bn,
24.4.28 En(12) =2-nEn.
24.4.29 B2n(13)=B2n(23)=-12(1-31-2n)B2n.
24.4.30 E2n-1(13)=-E2n-1(23)=-(1-31-2n)(22n-1)2nB2n,
n=1,2,.
24.4.31 Bn(14)=(-1)nBn(34)=-1-21-n2nBn-n4nEn-1,
n=1,2,.
24.4.32 B2n(16)=B2n(56)=12(1-21-2n)(1-31-2n)B2n,
24.4.33 E2n(16)=E2n(56)=1+3-2n22n+1E2n.

§24.4(vii) Derivatives

24.4.34 xBn(x) =nBn-1(x),
n=1,2,,
24.4.35 xEn(x) =nEn-1(x),
n=1,2,.

§24.4(viii) Symbolic Operations

Let P(x) denote any polynomial in x, and after expanding set (B(x))n=Bn(x) and (E(x))n=En(x). Then

24.4.36 P(B(x)+1)-P(B(x)) =P(x),
24.4.37 Bn(x+h) =(B(x)+h)n,
24.4.38 P(E(x)+1)+P(E(x)) =2P(x),
24.4.39 En(x+h) =(E(x)+h)n.

For these results and also connections with the umbral calculus see Gessel (2003).

§24.4(ix) Relations to Other Functions

For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).