About the Project
24 Bernoulli and Euler PolynomialsComputation

§24.19 Methods of Computation


§24.19(i) Bernoulli and Euler Numbers and Polynomials

Equations (24.5.3) and (24.5.4) enable Bn and En to be computed by recurrence. For higher values of n more efficient methods are available. For example, the tangent numbers Tn can be generated by simple recurrence relations obtained from (24.15.3), then (24.15.4) is applied. A similar method can be used for the Euler numbers based on (4.19.5). For details see Knuth and Buckholtz (1967).

Another method is based on the identities

24.19.1 N2n=2(2n)!(2π)2n(p-1|2np)(pp2np2n-1),
24.19.2 D2n =p-1|2np,
B2n =N2nD2n.

If N~2n denotes the right-hand side of (24.19.1) but with the second product taken only for p(πe)-12n+1, then N2n=N~2n for n2. For proofs and further information see Fillebrown (1992).

For other information see Chellali (1988) and Zhang and Jin (1996, pp. 1–11). For algorithms for computing Bn, En, Bn(x), and En(x) see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).

§24.19(ii) Values of Bn Modulo p

For number-theoretic applications it is important to compute B2n(modp) for 2np-3; in particular to find the irregular pairs (2n,p) for which B2n0(modp). We list here three methods, arranged in increasing order of efficiency.