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24 Bernoulli and Euler PolynomialsProperties

§24.15 Related Sequences of Numbers

Contents
  1. §24.15(i) Genocchi Numbers
  2. §24.15(ii) Tangent Numbers
  3. §24.15(iii) Stirling Numbers
  4. §24.15(iv) Fibonacci and Lucas Numbers

§24.15(i) Genocchi Numbers

24.15.1 2tet+1 =n=1Gntnn!,
24.15.2 Gn =2(12n)Bn.

See Table 24.15.1.

§24.15(ii) Tangent Numbers

24.15.3 tant=n=0Tntnn!,
24.15.4 T2n1=(1)n122n(22n1)2nB2n,
n=1,2,,
24.15.5 T2n=0,
n=0,1,.
Table 24.15.1: Genocchi and Tangent numbers.
n 0 1 2 3 4 5 6 7 8
Gn 0 1 1 0 1 0 3 0 17
Tn 0 1 0 2 0 16 0 272 0

§24.15(iii) Stirling Numbers

The Stirling numbers of the first kind s(n,m), and the second kind S(n,m), are as defined in §26.8(i).

24.15.6 Bn =k=0n(1)kk!S(n,k)k+1,
24.15.7 Bn =k=0n(1)k(n+1k+1)S(n+k,k)/(n+kk),
24.15.8 k=0n(1)n+ks(n+1,k+1)Bk=n!n+1.

In (24.15.9) and (24.15.10) p denotes a prime. See Horata (1991).

24.15.9 pBnnS(p1+n,p1)(modp2),
1np2,
24.15.10 2n14np2B2nS(p+2n,p1)(modp3),
22np3.

§24.15(iv) Fibonacci and Lucas Numbers

The Fibonacci numbers are defined by u0=0, u1=1, and un+1=un+un1, n1. The Lucas numbers are defined by v0=2, v1=1, and vn+1=vn+vn1, n1.

24.15.11 k=0n/2(n2k)(59)kB2kun2k =n6vn1+n3nv2n2,
24.15.12 k=0n/2(n2k)(54)kE2kvn2k =12n1.

For further information on the Fibonacci numbers see §26.11.