§24.19 Methods of Computation

Equations (24.5.3) and (24.5.4) enable $B_n$ and $E_n$ 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

If ${\tilde{N}}_{2n}$ denotes the right-hand side of (24.19.1) but with the second product taken only for $p\le \lfloor {(\pi \mathrm{e})}^{-1}2n\rfloor +1$, then $N_{2n}=\lceil {\tilde{N}}_{2n}\rceil$ for $n\ge 2$. 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).

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

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  Tanner and Wagstaff (1987) derives a congruence $(modp)$ 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		$$\frac{t^2}{\cosh t-1}=-2\sum _{n=0}^{\mathrm{\infty }}(2n-1)B_{2n}\frac{t^{2n}}{(2n)!},$$
  ⓘ Symbols: $B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $\cosh z$: 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 See also: Annotations for §24.19(ii), §24.19 and Ch.24

  and computes inverses modulo $p$ 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 $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).