Digital Library of Mathematical Functions
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24 Bernoulli and Euler PolynomialsProperties

§24.15 Related Sequences of Numbers

Contents

§24.15(i) Genocchi Numbers

24.15.1 2tt+1 =n=1Gntnn!,
24.15.2 Gn =2(1-2n)Bn.

See Table 24.15.1.

§24.15(ii) Tangent Numbers

24.15.3 tant=n=0Tntnn!,
24.15.4 T2n-1=(-1)n-122n(22n-1)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(p-1+n,p-1)(modp2),
1np-2,
24.15.10 2n-14np2B2nS(p+2n,p-1)(modp3),
22np-3.

§24.15(iv) Fibonacci and Lucas Numbers

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

24.15.11 k=0n/2(n2k)(59)kB2kun-2k =n6vn-1+n3nv2n-2,
24.15.12 k=0n/2(n2k)(54)kE2kvn-2k =12n-1.

For further information on the Fibonacci numbers see §26.11.