释义 |
Bernoulli NumberThere are two definitions for the Bernoulli numbers. The older one, no longer in widespread use, defines theBernoulli numbers by the equations
for , or
for (Whittaker and Watson 1990, p. 125). Gradshteyn and Ryzhik (1979) denote these numbers , whileBernoulli numbers defined by the newer (National Bureau of Standards) definition are denoted . The Bernoullinumbers 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 arethe 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
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 inthe 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 curiousproperty is that the fraction part of in Decimal has a Decimal Period which divides , and there isa single digit before that period (Conway 1996).
(Sloane's A000367and A002445). In addition,
 | (22) |
for , 2, ....
Bernoulli first used the Bernoulli numbers while computing . He used the property of the FigurateNumber 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 theform 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). Plouffeand 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 Computationof 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. SequencesA000367/M4039and A002445/M4189in ``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.
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