[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 , 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  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  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.