§24.19 Methods of Computation

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

§24.19(i) Bernoulli and Euler Numbers and Polynomials
§24.19(ii) Values of $\mathop{B_{{n}}\/}\nolimits$ Modulo $p$

§24.19(i) Bernoulli and Euler Numbers and Polynomials

Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.19.i

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

24.19.1 $N_{{2n}}=\frac{2(2n)!}{(2\pi)^{{2n}}}\left(\prod _{{p-1\divides 2n}}p\right)\left(\prod _{p}\frac{p^{{2n}}}{p^{{2n}}-1}\right),$

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

$D_{{2n}}=\prod _{{p-1\divides 2n}}p,$

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

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $n$ : integer and $p$ : prime
Permalink:: http://dlmf.nist.gov/24.19.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

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 $p$

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.

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

24.19.3 $\frac{t^{2}}{\mathop{\cosh\/}\nolimits t-1}=-2\sum _{{n=0}}^{{\infty}}(2n-1)\mathop{B_{{2n}}\/}\nolimits\frac{t^{{2n}}}{(2n)!},$

Symbols:

$\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $!$ : $n!$ : factorial, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $n$ : integer and $t$ : real or complex

Referenced by:

§24.19(ii)

Permalink:

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

Encodings:

TeX, pMML, png

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).
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 $(2n,p)$ . Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

§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 $\mathop{B_{{n}}\/}\nolimits$ Modulo $p$