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Research Article

Open Access Peer-reviewed

F. M. S. Lima^{ }

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

In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers *B*_{2}_{n }, *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*_{2}_{n}. Interestingly, it is suitable for large values of *n* since it allows the computation of both *B*_{4}_{n} and *B*_{4}_{n}_{+2} from only *B*_{0}, *B*_{2}, ..., *B*_{2}_{n}.

The Bernoulli numbers *B*_{n}, *n* being a nonnegative integer, can be defined by the generating function

(1) |

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

The numbers *B*_{n} 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 *B*_{n} 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 *e*^{x}-1, using the Cauchy product with the Maclaurin series for *e*^{x}-1, and equating the coefficients, which yields

(2) |

This kind of recurrence has the disadvantage of demanding the previous knowledge of *every* *B*_{0}, *B*_{1}, ...,*B*_{n}_{-1} for the computation of *B*_{n}. 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 *B*_{n} is determined only from every second, or every third, etc., preceding *B*_{n} (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 *B*_{n}_{-1} in order to compute *B*_{n} ^{ 4}. For an extensive study of these and other recurrence relations involving the numbers *B*_{n}, see ^{ 5}.

Here in this note, I apply the Euler's formula relating the even zeta value (2*n*) to *B*_{2}_{n} to a recurrence formula for (2*n*) obtained by Kuo in ^{ 6} in order to convert it into a new recurrence formula for *B*_{2}_{n}. This reveals a third type of recurrence for the Bernoulli numbers, in the sense that it allows us to compute both *B*_{4}_{n} and *B*_{4}_{n}_{+2} from only the *first-half* of the preceding numbers, i.e. *B*_{0}, *B*_{2}, ..., *B*_{2}_{n}.

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}:

(3) |

Since (-1)^{n}^{-1} *B*_{2}_{n} is a positive rational for all integer *n*>0, then (2*n*) is a positive rational multiple of ^{2}^{n}. 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}:

(4) |

valid for all complex *s* 1 +(2*m*)*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 ****B**_{2}_{n}**). **For any integer *n*>0, the following formula holds:

where

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

(5) |

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

(6) |

Now, by factoring (-1)^{n}^{-1} one finds

(7) |

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

For instance, when *n*=1, since (1)_{1} = 1 and *B*_{0}=1, one readily finds

(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 *B*_{2}_{n} 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 *B*_{2}_{n}, 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.

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

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

[1] | I.S. Gradshteyn and I.M. Rhyzik, Table of Integrals, Series, and Products, 7th ed., Academic Press, New York, 2007. | ||

In article | View Article | ||

[2] | J. Worpitzky, “Studien über die Bernoullischen und Eulerschen Zahlen,” J. Reine Angew. Math. 94, 203-232 (1883). | ||

In article | View Article | ||

[3] | S. Ramanujan, “Some Properties of Bernoulli's Numbers,” J. Indian Math. Soc. 3, 219-234 (1911). | ||

In article | View Article | ||

[4] | T. Agoh and K. Dilcher, “Shortened recurrence relations for Bernoulli numbers,” Discrete Math. 309, 887-898 (2009). | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[6] | H.-T. Kuo, “A recurrence formula for (2n),” Bull. Am. Math. Soc. 55, 573-574 (1949). | ||

In article | View Article | ||

[7] | L. Euler, “De summis serierum reciprocarum,” Commentarii Academiae Scientiarum Petropolitanae 7, 123-134 (1740). | ||

In article | |||

[8] | E.C. Titchmarsh, The theory of the Riemann Zeta-function, 2nd ed., Clarendon Press, Oxford, 1986. | ||

In article | View Article | ||

[9] | H. Hasse, “Ein Summierungsverfahren für die Riemannsche Zeta-Reihe,” Math. Z. 32, 458-464 (1930). | ||

In article | View Article | ||

[10] | S. Akiyama and Y. Tanigawa, “Multiple Zeta Values at Non-Positive Integers,” Ramanujan J. 5, 327-351 (2001). | ||

In article | View Article | ||

[11] | R.P. Brent and P. Zimmermann, Modern computer arithmetic, Cambridge University Press, New York, 2010. Sec. 4.7.2. | ||

In article | View Article | ||

[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). | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2018 F. M. S. Lima

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/

F. M. S. Lima. A Shortened Recurrence Relation for Bernoulli Numbers. *Turkish Journal of Analysis and Number Theory*. Vol. 6, No. 2, 2018, pp 49-51. http://pubs.sciepub.com/tjant/6/2/3

Lima, F. M. S.. "A Shortened Recurrence Relation for Bernoulli Numbers." *Turkish Journal of Analysis and Number Theory* 6.2 (2018): 49-51.

Lima, F. M. S. (2018). A Shortened Recurrence Relation for Bernoulli Numbers. *Turkish Journal of Analysis and Number Theory*, *6*(2), 49-51.

Lima, F. M. S.. "A Shortened Recurrence Relation for Bernoulli Numbers." *Turkish Journal of Analysis and Number Theory* 6, no. 2 (2018): 49-51.

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[1] | I.S. Gradshteyn and I.M. Rhyzik, Table of Integrals, Series, and Products, 7th ed., Academic Press, New York, 2007. | ||

In article | View Article | ||

[2] | J. Worpitzky, “Studien über die Bernoullischen und Eulerschen Zahlen,” J. Reine Angew. Math. 94, 203-232 (1883). | ||

In article | View Article | ||

[3] | S. Ramanujan, “Some Properties of Bernoulli's Numbers,” J. Indian Math. Soc. 3, 219-234 (1911). | ||

In article | View Article | ||

[4] | T. Agoh and K. Dilcher, “Shortened recurrence relations for Bernoulli numbers,” Discrete Math. 309, 887-898 (2009). | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[6] | H.-T. Kuo, “A recurrence formula for (2n),” Bull. Am. Math. Soc. 55, 573-574 (1949). | ||

In article | View Article | ||

[7] | L. Euler, “De summis serierum reciprocarum,” Commentarii Academiae Scientiarum Petropolitanae 7, 123-134 (1740). | ||

In article | |||

[8] | E.C. Titchmarsh, The theory of the Riemann Zeta-function, 2nd ed., Clarendon Press, Oxford, 1986. | ||

In article | View Article | ||

[9] | H. Hasse, “Ein Summierungsverfahren für die Riemannsche Zeta-Reihe,” Math. Z. 32, 458-464 (1930). | ||

In article | View Article | ||

[10] | S. Akiyama and Y. Tanigawa, “Multiple Zeta Values at Non-Positive Integers,” Ramanujan J. 5, 327-351 (2001). | ||

In article | View Article | ||

[11] | R.P. Brent and P. Zimmermann, Modern computer arithmetic, Cambridge University Press, New York, 2010. Sec. 4.7.2. | ||

In article | View Article | ||

[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). | ||

In article | View Article | ||