Bernoulli number


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Bernoulli number

[ber‚nü·lē ‚nəm·bər]
(mathematics)
The numerical value of the coefficient of x 2n /(2 n)! is the expansion of xex /(ex-1).
References in periodicals archive ?
[B.sub.n], [B.sub.n](x) are the Bernoulli number and Bernoulli polynomial, respectively.
Let, as usual, [[gamma].sub.0] denote Euler's constant, and [B.sub.j] stand for the j-th Bernoulli number. Then the following theorem is true.
In this expression, [B.sub.i] is a Bernoulli number. The sum converges for all A > 2, and one obtains an error in F(A) estimated to be as small as [10.sup.-14] for A > 4 if one sums up to s = 28.
where [B.sub.j] denotes a standard Bernoulli number.
It is interesting to note that there are already classical formulas expressing the Bernoulli number in terms of Stirling numbers such as
Kim, An explicit formula on the generalized Bernoulli number with order n, Indian J.
We call [[beta].sub.n,q] Bernoulli number and [[beta].sub.n,q] = [[beta].sub.n,q](0).
where [B.sub.k] := [B.sub.k](0) is the k-th Bernoulli number and [E.sub.k] := [E.sub.k](1) is the k-th Euler number.
where [B.sub.j] is the j-th Bernoulli number, [alpha] > 0 and [beta] > 0 satisfy [alpha][beta] = [[pi].sup.2], and [summation]' means that, when k is an odd number 2m - 1, the last term of the left hand side in (8) is taken as [(-1).sup.m][[pi].sup.2m][B.sup.2.sub.2m]/[(m!).sup.2].
In [4], Arakawa and Kaneko used analytic continuation of [zeta]([s.sub.1], ..., [s.sub.d]) as a function of one variable [s.sub.d] when [s.sub.1], ..., [s.sub.d-1] are positive integers and discussed the relation among generalized Bernoulli numbers. For a general d, Zhao [12] proved the analytic continuation of [zeta]([s.sub.1], ..., [s.sub.d]) as a function of d variables using the theory of generalized function and Akiyama, Egami and Tanigawa [1] proved the same result by applying the classical Euler-Maclaurin formula to the index of the summation [n.sub.d].
where [[alpha].sub.i], i = 0, 1, ..., m are the Bernoulli numbers. Thus, the first four such polynomials, respectively, are
For the second edition he has corrected errors; clarified some confusion; and added new problems on Bernoulli numbers, metric spaces, and differential equations.