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Computation of tangent, Euler, and Bernoulli numbers. (English) Zbl 0178.04401
The tangent numbers $T_n$, Euler numbers $E_n$, and Bernoulli numbers $B_n$ are defined to be the coefficients in the following power series $$\tan z=\sum_{n\ge 0} T_nz^n/z!,\qquad \sec z=\sum_{n\ge 0} E_nz^n/z!,\qquad z/(e^z-1)=\sum_{n\ge 0} B_nz^n/z!.$$ The authors discuss traditional recurrence relations used for calculating these numbers, and then proceed to develop calculating techniques more suitable for modern binary computers based on new recurrence relations which are derived through simple trigonometric identities. Two theorems are given about the periodicity of $T_n$ and $E_n$ modulo an odd prime $p$. A table is given for the integers $T_n$ and $E_n$ $(n\le 120)$ and for the numbers $B_n$ $(n\le 250)$. The $B_n$ ar given in the Staudt-Clausen form $B_{2n}=C_{2n}-\sum 1/p$ where the $C_{2n}$ are integers and the summation runs over all primes $p$ such that $(p-1)$ divides $2n$. The authors have deposited a copy of the values of $T_n$ $(n\le 835)$, $E_n$ $(n\le 808)$ and $B_n$ $(n\le 836)$ in the Unpublished Mathematical Tables of Math. Comput.
Reviewer: T. R. Parkin

MSC:
11Y16Algorithms; complexity (number theory)
11B68Bernoulli and Euler numbers and polynomials
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