Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions

The \emph{Barnes $\zeta$-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 $\Re(x) > 0$ and $\Re(z) > n$ and continued meromorphically to \C. Specialized at negative integers $-k$, the Barnes $\zeta$-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 $\zeta$-functions, Bernoulli--Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.Comment: 11 page