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A note on Bernoulli numbers and Shintani generalized Bernoulli polynomials. (English) Zbl 0864.11043

Generalized Bernoulli polynomials \(B_m(A,x)\) were introduced by Shintani in 1976 in order to express the special values at nonpositive integers of Dedekind zeta functions for totally real number fields: For any \(r\times n\) matrix \(A= (a_{jk})\) with positive entries and any \(r\)-tuple of complex numbers \(x=(x_1, \dots, x_r)\) let the zeta function \(\zeta (s,A,x)\) be defined by \[ \zeta(s,A,x)= \sum_{n_1\geq 0} \cdots \sum_{n_r\geq 0} \prod^n_{k=1} \bigl[a_{1,k} (n_1+x_1)+ \cdots+ a_{rk} (n_r+x_r) \bigr]^{-s}, \] then \(\zeta (1-m,A,x) =(-1)^r m^{-n} B_m (A,x)\). Although it is possible to express \(B_m(A,x)\) in terms of a combination of products of ordinary Bernoulli polynomials it is quite painful and laborious to compute them explicitly.
In the note under review the author determines \(\zeta (1-m,A,x)\) by a finite set of polynomials which can be obtained by integrating over certain simplexes. As a consequence, he gives some examples of identities among the ordinary Bernoulli polynomials which are difficult to prove otherwise.

MSC:

11M41 Other Dirichlet series and zeta functions
11B68 Bernoulli and Euler numbers and polynomials
11R42 Zeta functions and \(L\)-functions of number fields
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