## Bernoulli Number

There are two definitions for the Bernoulli numbers. The older one, no longer in widespread use, defines the Bernoulli numbers by the equations

 (1)

for , or
 (2) (3)

for (Whittaker and Watson 1990, p. 125). Gradshteyn and Ryzhik (1979) denote these numbers , while Bernoulli numbers defined by the newer (National Bureau of Standards) definition are denoted . The Bernoulli numbers may be calculated from the integral
 (4)

and analytically from
 (5)

for , 2, ..., where is the Riemann Zeta Function.

The first few Bernoulli numbers are

Bernoulli numbers defined by the modern definition are denoted and also called Even-index'' Bernoulli numbers. These are the Bernoulli numbers returned by the Mathematica (Wolfram Research, Champaign, IL) function BernoulliB[n]. These Bernoulli numbers are a superset of the archaic ones since

 (6)

The can be defined by the identity
 (7)

These relationships can be derived using the generating function
 (8)

which converges uniformly for and all (Castellanos 1988). Taking the partial derivative gives
 (9)

The solution to this differential equation is
 (10)

so integrating gives
 (11)

or
 (12)

(Castellanos 1988). Setting and adding to both sides then gives
 (13)

Letting then gives
 (14)

for . The Bernoulli numbers may also be calculated from the integral
 (15)

or from
 (16)

The Bernoulli numbers satisfy the identity

 (17)

where is a Binomial Coefficient. An asymptotic Formula is
 (18)

Bernoulli numbers appear in expressions of the form , where , 2, .... Bernoulli numbers also appear in the series expansions of functions involving , , , , , , , , and . An analytic solution exists for Even orders,
 (19)

for , 2, ..., where is the Riemann Zeta Function. Another intimate connection with the Riemann Zeta Function is provided by the identity
 (20)

The Denominator of is given by the von Staudt-Clausen Theorem

 (21)

which also implies that the Denominator of is Squarefree (Hardy and Wright 1979). Another curious property is that the fraction part of in Decimal has a Decimal Period which divides , and there is a single digit before that period (Conway 1996).

(Sloane's A000367 and A002445). In addition,
 (22)

for , 2, ....

Bernoulli first used the Bernoulli numbers while computing . He used the property of the Figurate Number Triangle that

 (23)

along with a form for which he derived inductively to compute the sums up to (Boyer 1968, p. 85). For , the sum is given by
 (24)

where the Notation means the quantity in question is raised to the appropriate Power , and all terms of the form are replaced with the corresponding Bernoulli numbers . Written explicitly in terms of a sum of Powers,
 (25)

It is also true that the Coefficients of the terms in such an expansion sum to 1 (which Bernoulli stated without proof). Ramanujan gave a number of curious infinite sum identities involving Bernoulli numbers (Berndt 1994).

G. J. Fee and S. Plouffe have computed , which has Digits (Plouffe). Plouffe and collaborators have also calculated for up to 72,000.

See also Argoh's Conjecture, Bernoulli Function, Bernoulli Polynomial, Debye Functions, Euler-Maclaurin Integration Formulas, Euler Number, Figurate Number Triangle, Genocchi Number, Pascal's Triangle, Riemann Zeta Function, von Staudt-Clausen Theorem

References

Abramowitz, M. and Stegun, C. A. (Eds.). Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula.'' §23.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804-806, 1972.

Arfken, G. Bernoulli Numbers, Euler-Maclaurin Formula.'' §5.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 327-338, 1985.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 71, 1987.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 81-85, 1994.

Boyer, C. B. A History of Mathematics. New York: Wiley, 1968.

Castellanos, D. The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988.

Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 107-110, 1996.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1980.

Hardy, G. H. and Wright, W. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 91-93, 1979.

Ireland, K. and Rosen, M. Bernoulli Numbers.'' Ch. 15 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 228-248, 1990.

Knuth, D. E. and Buckholtz, T. J. Computation of Tangent, Euler, and Bernoulli Numbers.'' Math. Comput. 21, 663-688, 1967.

Plouffe, S. Plouffe's Inverter: Table of Current Records for the Computation of Constants.'' http://www.lacim.uqam.ca/pi/records.html.

Ramanujan, S. Some Properties of Bernoulli's Numbers.'' J. Indian Math. Soc. 3, 219-234, 1911.

Sloane, N. J. A. Sequences A000367/M4039 and A002445/M4189 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Spanier, J. and Oldham, K. B. The Bernoulli Numbers, .'' Ch. 4 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 35-38, 1987.

Wagstaff, S. S. Jr. Ramanujan's Paper on Bernoulli Numbers.'' J. Indian Math. Soc. 45, 49-65, 1981.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.