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24 Bernoulli and Euler PolynomialsComputation

§24.19 Methods of Computation

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§24.19(i) Bernoulli and Euler Numbers and Polynomials

Equations (24.5.3) and (24.5.4) enable \mathop{B_{{n}}\/}\nolimits and \mathop{E_{{n}}\/}\nolimits to be computed by recurrence. For higher values of n more efficient methods are available. For example, the tangent numbers T_{n} 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

24.19.1N_{{2n}}=\frac{2(2n)!}{(2\pi)^{{2n}}}\left(\prod_{{p-1\divides 2n}}p\right)% \left(\prod_{p}\frac{p^{{2n}}}{p^{{2n}}-1}\right),
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Symbols:
!: n!: factorial, n: integer and p: prime
Referenced by:
§24.19(i)
Permalink:
http://dlmf.nist.gov/24.19.E1
Encodings:
TeX, pMML, png
24.19.2
D_{{2n}}=\prod_{{p-1\divides 2n}}p,
\mathop{B_{{2n}}\/}\nolimits=\dfrac{N_{{2n}}}{D_{{2n}}}.

If \widetilde{N}_{{2n}} denotes the right-hand side of (24.19.1) but with the second product taken only for p\leq\left\lfloor(\pi e)^{{-1}}2n\right\rfloor+1, then N_{{2n}}=\left\lceil\widetilde{N}_{{2n}}\right\rceil for n\geq 2. 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 \mathop{B_{{n}}\/}\nolimits, \mathop{E_{{n}}\/}\nolimits, \mathop{B_{{n}}\/}\nolimits\!\left(x\right), and \mathop{E_{{n}}\/}\nolimits\!\left(x\right) see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).

§24.19(ii) Values of \mathop{B_{{n}}\/}\nolimits Modulo p

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Notes:
(24.19.3) follows from (24.2.1) by series manipulations.
Keywords:
Bernoulli numbers, Stickelberger codes
Permalink:
http://dlmf.nist.gov/24.19.ii

For number-theoretic applications it is important to compute \mathop{B_{{2n}}\/}\nolimits\;\;(\mathop{{\rm mod}}p) for 2n\leq p-3; in particular to find the irregular pairs (2n,p) for which \mathop{B_{{2n}}\/}\nolimits\equiv 0\;\;(\mathop{{\rm mod}}p). We list here three methods, arranged in increasing order of efficiency.

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