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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×n matrix A=(a jk ) with positive entries and any r-tuple of complex numbers x=(x 1 ,,x r ) let the zeta function ζ(s,A,x) be defined by

ζ(s,A,x)= n 1 0 n r 0 k=1 n a 1,k (n 1 +x 1 ) + + a rk (n r +x r ) -s ,

then ζ(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 ζ(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:
11M41Other Dirichlet series and zeta functions
11B68Bernoulli and Euler numbers and polynomials
11R42Zeta functions and L-functions of global number fields