[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.