Generalized Bernoulli polynomials 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 matrix with positive entries and any -tuple of complex numbers let the zeta function be defined by
then . Although it is possible to express 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 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.