zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $$\zeta(s,A,x)= \sum_{n_1\ge 0} \cdots \sum_{n_r\ge 0} \prod^n_{k=1} \bigl[a_{1,k} (n_1+x_1)+ \cdots+ a_{rk} (n_r+x_r) \bigr]^{-s},$$ 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.

11M41Other Dirichlet series and zeta functions
11B68Bernoulli and Euler numbers and polynomials
11R42Zeta functions and $L$-functions of global number fields
Full Text: DOI