Computation of tangent, Euler, and Bernoulli numbers
Authors:
Donald E. Knuth and Thomas J. Buckholtz
Journal:
Math. Comp. 21 (1967), 663688
MSC:
Primary 65.25
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. 351352.
 [2]
S. A. Joffe, "Calculation of the first thirtytwo Eulerian numbers from central differences of zero," Quart. J. Math., v. 47, 1916, pp. 103126.
 [3]
S. A. Joffe, "Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero," Quart. J. Math., v. 48, 19171920, pp. 193271.
 [4]
D.
H. Lehmer, An extension of the table of Bernoulli numbers,
Duke Math. J. 2 (1936), no. 3, 460–464. MR
1545938, http://dx.doi.org/10.1215/S0012709436002387
 [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. 18.
 [8]
K. G. C. von Staudt, "Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend," J. für Math., v. 21, 1840, pp. 372374.
 [1]
 Thomas Clausen, "Theorem," Astr. Nachrichten, v. 17, 1840, cols. 351352.
 [2]
 S. A. Joffe, "Calculation of the first thirtytwo Eulerian numbers from central differences of zero," Quart. J. Math., v. 47, 1916, pp. 103126.
 [3]
 S. A. Joffe, "Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero," Quart. J. Math., v. 48, 19171920, pp. 193271.
 [4]
 D. H. Lehmer, "An extension of the table of Bernoulli numbers," Duke Math. J., v. 2, 1936, pp. 460464. MR 1545938
 [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. 18.
 [8]
 K. G. C. von Staudt, "Beweis eines Lehrsatzes die Bernoullischen Zahlen betreffend," J. für Math., v. 21, 1840, pp. 372374.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718196702217359
PII:
S 00255718(1967)02217359
Article copyright:
© Copyright 1967
American Mathematical Society
