24 Bernoulli and Euler Polynomials Computation 24.18 Physical Applications 24.20 Tables

§24.19 Methods of Computation

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Permalink:: http://dlmf.nist.gov/24.19
See also:: Annotations for Ch.24

§24.19(i) Bernoulli and Euler Numbers and Polynomials
§24.19(ii) Values of $B_{n}$ Modulo

§24.19(i) Bernoulli and Euler Numbers and Polynomials

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, computation
Permalink:: http://dlmf.nist.gov/24.19.i
See also:: Annotations for §24.19 and Ch.24

Equations (24.5.3) and (24.5.4) enable $B_{n}$ and $E_{n}$ to be computed by recurrence. For higher values of 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

24.19.1 $N_{2n}=\frac{2(2n)!}{(2\pi)^{2n}}\left(\prod_{p-1\mathbin{|}2n}p\right)\left(% \prod_{p}\frac{p^{2n}}{p^{2n}-1}\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : integer and : prime
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.19.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §24.19(i), §24.19 and Ch.24

24.19.2

$D_{2n}=\prod_{p-1\mathbin{|}2n}p,$

$B_{2n}=\dfrac{N_{2n}}{D_{2n}}.$

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, : integer and : prime
Permalink:: http://dlmf.nist.gov/24.19.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §24.19(i), §24.19 and Ch.24

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 $B_{n}$ , $E_{n}$ , $B_{n}\left(x\right)$ , and $E_{n}\left(x\right)$ see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180).

§24.19(ii) Values of $B_{n}$ Modulo

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Keywords:: Bernoulli numbers, Stickelberger codes, irregular pairs
Notes:: (24.19.3) follows from (24.2.1) by series manipulations.
Referenced by:: Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/24.19.ii
See also:: Annotations for §24.19 and Ch.24

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

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Tanner and Wagstaff (1987) derives a congruence $\pmod{p}$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).
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Buhler et al. (1992) uses the expansion

24.19.3 $\frac{t^{2}}{\cosh t-1}=-2\sum_{n=0}^{\infty}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},$
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Symbols:

$B_{\NVar{n}}$ : Bernoulli numbers, : factorial (as in ), $\cosh\NVar{z}$ : hyperbolic cosine function, : integer and : real or complex

Referenced by:

§24.19(ii)

Permalink:

http://dlmf.nist.gov/24.19.E3

Encodings:

TeX, pMML, png

See also:

Annotations for §24.19(ii), §24.19 and Ch.24

and computes inverses modulo of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).
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A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

24.18 Physical Applications 24.20 Tables

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§24.19 Methods of Computation

Contents

§24.19(i) Bernoulli and Euler Numbers and Polynomials

§24.19(ii) Values of Modulo

§24.19(ii) Values of $B_{n}$ Modulo