A Shortened Recurrence Relation for Bernoulli Numbers

F. M. S. Lima^1,

¹Institute of Physics, University of Brasília, P.O. Box 04455, 70919-970, Brasília-DF, Brazil

Cite this paper:
F. M. S. Lima. A Shortened Recurrence Relation for Bernoulli Numbers. Turkish Journal of Analysis and Number Theory. 2018; 6(2):49-51. doi: 10.12691/tjant-6-2-3

Abstract

In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers B_2n , n being a positive integer. This formula is shown to be advantageous in comparison to other known formulae for the exact symbolic computation of B_2n. Interestingly, it is suitable for large values of n since it allows the computation of both B_4n and B_4n+2 from only B₀, B₂, ..., B_2n.

Keywords:
Bernoulli numbers recurrence relations Riemann zeta function

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	I.S. Gradshteyn and I.M. Rhyzik, Table of Integrals, Series, and Products, 7th ed., Academic Press, New York, 2007.
[2]	J. Worpitzky, “Studien über die Bernoullischen und Eulerschen Zahlen,” J. Reine Angew. Math. 94, 203-232 (1883).
[3]	S. Ramanujan, “Some Properties of Bernoulli's Numbers,” J. Indian Math. Soc. 3, 219-234 (1911).
[4]	T. Agoh and K. Dilcher, “Shortened recurrence relations for Bernoulli numbers,” Discrete Math. 309, 887-898 (2009).
[5]	K. Dilcher, L. Skula, I.Sh. Slavutskii, “Bernoulli numbers. Bibliography (1713-1990),” Queen's Papers in Pure and Applied Mathematics 87, Queen's University, Kingston, Ont., 1991. Updated on-line version: http://www.mathstat.dal.ca/~dilcher/bernoulli.html.
[6]	H.-T. Kuo, “A recurrence formula for (2n),” Bull. Am. Math. Soc. 55, 573-574 (1949).
[7]	L. Euler, “De summis serierum reciprocarum,” Commentarii Academiae Scientiarum Petropolitanae 7, 123-134 (1740).
[8]	E.C. Titchmarsh, The theory of the Riemann Zeta-function, 2nd ed., Clarendon Press, Oxford, 1986.
[9]	H. Hasse, “Ein Summierungsverfahren für die Riemannsche Zeta-Reihe,” Math. Z. 32, 458-464 (1930).
[10]	S. Akiyama and Y. Tanigawa, “Multiple Zeta Values at Non-Positive Integers,” Ramanujan J. 5, 327-351 (2001).
[11]	R.P. Brent and P. Zimmermann, Modern computer arithmetic, Cambridge University Press, New York, 2010. Sec. 4.7.2.
[12]	R.P. Brent and D. Harvey, “Fast computation of Bernoulli, Tangent and Secant numbers.” Springer Proceedings in Mathematics and Statistics, vol. 50, 127-142 (2013).