Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

A Shortened Recurrence Relation for Bernoulli Numbers

F. M. S. Lima

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

Abstract: In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers B₂n , n being a positive integer. This formula is shown to be advantageous in comparison to other known formulae for the exact symbolic computation of B₂n. Interestingly, it is suitable for large values of n since it allows the computation of both B₄n and B₄n₊₂ from only B₀, B₂, ..., B₂n.

Keywords: Bernoulli numbers, recurrence relations, Riemann zeta function

Received January 13, 2018; Revised March 03, 2018; Accepted April 09, 2018

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

1. Introduction

The Bernoulli numbers Bn, n being a nonnegative integer, can be defined by the generating function

[picture] | (1)

which holds for all complex z with |z| < 2. The first few values are B₀ = 1, B₁ = -1/2, and B₂ = 1/6. It is well-known that Bn = 0 for odd values of n, n>1. For even values of n, n>1, the numbers Bn form a subsequence of non-null rational numbers such that (-1)m⁺¹ B₂m>0 for all positive integer m. In other words, the entire subsequence of non-null Bernoulli numbers up to any Bn, n>1, consists of B₀ and B₁, given above, and the preceding numbers B₂m, m = 1, ..., n/2. The basic properties of these numbers can be found, e.g., in Sec. 9.61 of [1].

The numbers Bn appear in many instances in pure and applied mathematics, most notably in number theory, finite differences calculations, and asymptotic analysis. They also appear as coefficients in the Euler-Maclaurin formula, thus being important for accurate numerical evaluation of special functions. Therefore, numerical computations of Bn are of great mathematical interest. To this end, recurrence formulae were soon recognized as the most efficient tool [2]. One of the simplest such formulas is found by multiplying both sides of Eq. (1) by ex-1, using the Cauchy product with the Maclaurin series for ex-1, and equating the coefficients, which yields

[picture] | (2)

This kind of recurrence has the disadvantage of demanding the previous knowledge of every B₀, B₁, ...,Bn₋₁ for the computation of Bn. On searching for more efficient formulae in literature, one finds shortened recurrence relations of two different types. The first type consists of lacunary recurrences, in which Bn is determined only from every second, or every third, etc., preceding Bn (see, e.g., the lacunary formula by Ramanujan in [3]). The second type demands the knowledge of the second-half of Bernoulli numbers up to Bn₋₁ in order to compute Bn [4]. For an extensive study of these and other recurrence relations involving the numbers Bn, see [5].

Here in this note, I apply the Euler's formula relating the even zeta value (2n) to B₂n to a recurrence formula for (2n) obtained by Kuo in [6] in order to convert it into a new recurrence formula for B₂n. This reveals a third type of recurrence for the Bernoulli numbers, in the sense that it allows us to compute both B₄n and B₄n₊₂ from only the first-half of the preceding numbers, i.e. B₀, B₂, ..., B₂n.

2. Recurrence Relation for Bernoulli Numbers

For complex values of s with Re(s)>1, the Riemann zeta function is defined as In this domain, the convergence of this series is guaranteed by the integral test. For s=1, one has the well-known harmonic serie , which diverges to infinity. For positive even values of s, one has a celebrated formula by Euler [7]:

[picture] | (3)

Since (-1)n⁻¹ B₂n is a positive rational for all integer n>0, then (2n) is a positive rational multiple of ²n. The function (s), as defined above, can be extended to the entire complex plane (except for the only simple pole at s=1) by analytic continuation, which yields (0) = -1/2 [8]. Alternatively, we can use a globally convergent series representation due to Hasse (1930) [9]:

[picture] | (4)

valid for all complex s 1 +(2m)i/ln(2), m being an integer. At s=0, it reduces to (0) = -1/2, which shows that Eq. (3) is also valid for n=0.

These are the ingredients we need to state our first lemma, which comes from a little-known recurrence formula by Kuo (1949) [6], written here in terms of even zeta values.

Lemma 1 (Kuo's recurrence formula). For any positive integer n, one has

This formula is proved in [6] by developing successive integrations (from 0 to x) of the Fourier series , which converges to (-x)/2 for all x (0,2), and then applying Parseval's theorem.

We are now in a position to state and prove our main result. In what follows, (λ)n:= λ (λ+1) (λ+n-1) is the Pochhammer symbol.

Theorem 1 (Recurrence for B₂n). For any integer n>0, the following formula holds:

where

Proof. On dividing both sides of Kuo's formula, our Lemma 1, by (2)²n, one finds

[picture] | (5)

From Eq. (3), we know that , which is valid for all integer m 0. By substituting this in Eq. (5), one finds

[picture] | (6)

Now, by factoring (-1)n⁻¹ one finds

[picture] | (7)

On writing the first term in the right-hand side as (2n)!/[4(2n-1)(n-1)!²] and putting (2n)!/(n-1)! in evidence (out of the square brackets), the desired result follows.

For instance, when n=1, since (1)₁ = 1 and B₀=1, one readily finds

[picture] | (8)

For larger values of n, a more efficient, fast computation is achieved when one avoids the repeated computation of the factorials in our Theorem 1. For this, it is advantageous to make use of the algorithm given below. A short Maple code implementation of this algorithm is freely available in the journal website (file B4mB4mp2.mws).

Of course, more efficient numerical routines for floating-point computation of B₂n can be found in some recent works (see, e.g., [10,11,12]), but we are interested here only in exact symbolic computations of the rational numbers B₂n, which are important for implementation in modern mathematical software such as Maple and Mathematica. Admittedly, most algorithms in [12] can, a priori, be implemented in a manner to yield exact symbolic results.

2.1. Algorithm

For completion, the algorithm corresponding to an efficient implementation of the recurrence formula given in our Theorem 1 is presented below.

Acknowledgements

The author wishes to thank M. R. Javier for checking computationally all results in this work.

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).