2 research outputs found
Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions
The \emph{Barnes -function} is
\zeta_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 +
\dots + m_n a_n \right)^z} defined for and and
continued meromorphically to \C. Specialized at negative integers , the
Barnes -function gives
\zeta_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) where
B_k(x; \a) is a \emph{Bernoulli--Barnes polynomial}, which can be also
defined through a generating function that has a slightly more general form
than that for Bernoulli polynomials. Specializing B_k(0; \a) gives the
\emph{Bernoulli--Barnes numbers}. We exhibit relations among Barnes
-functions, Bernoulli--Barnes numbers and polynomials, which generalize
various identities of Agoh, Apostol, Dilcher, and Euler.Comment: 11 page