Computation of tangent, Euler, and Bernoulli numbers

Authors:
Donald E. Knuth and Thomas J. Buckholtz

Journal:
Math. Comp. **21** (1967), 663-688

MSC:
Primary 65.25

DOI:
https://doi.org/10.1090/S0025-5718-1967-0221735-9

MathSciNet review:
0221735

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Abstract: Some elementary methods are described which may be used to calculate tangent numbers, Euler numbers, and Bernoulli numbers much more easily and rapidly on electronic computers than the traditional recurrence relations which have been used for over a century. These methods have been used to prepare an accompanying table which extends the existing tables of these numbers. Some theorems about the periodicity of the tangent numbers, which were suggested by the tables, are also proved.

**[1]**Thomas Clausen, "Theorem,"*Astr. Nachrichten*, v. 17, 1840, cols. 351-352.**[2]**S. A. Joffe, "Calculation of the first thirty-two Eulerian numbers from central differences of zero,"*Quart. J. Math.*, v. 47, 1916, pp. 103-126.**[3]**S. A. Joffe, "Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero,"*Quart. J. Math.*, v. 48, 1917-1920, pp. 193-271.**[4]**D. H. Lehmer,*An extension of the table of Bernoulli numbers*, Duke Math. J.**2**(1936), no. 3, 460–464. MR**1545938**, https://doi.org/10.1215/S0012-7094-36-00238-7**[5]**Niels Nielsen, Traité Élémentaire des Nombres de Bernoulli, Paris, 1923.**[6]**J. Peters & J. Stein, Zehnstellige Logarithmentafel, Berlin, 1922.**[7]**S. Z. Serebrennikoff, "Tables des premiers quatre vingt dix nombres de Bernoulli,"*Mém. Acad. St. Petersbourg*8, v. 16, 1905, no. 10, pp. 1-8.**[8]**K. G. C. von Staudt, "Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend,"*J. für Math.*, v. 21, 1840, pp. 372-374.

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DOI:
https://doi.org/10.1090/S0025-5718-1967-0221735-9

Article copyright:
© Copyright 1967
American Mathematical Society