3 Relations for Bernoulli–Barnes Numbers and Barnes Zeta Functions (original) (raw)

The Barnes ζ-function is ζ n (z, x; a) := ∑ m∈Z n ≥0 1 (x + m 1 a 1 + • • • + m n a n) z defined for Re(x) > 0 and Re(z) > n and continued meromorphically to C. Specialized at negative integers −k, the Barnes ζ-function gives ζ n (−k, x; a) = (−1) n k! (k + n)! B k+n (x; a) where B k (x; a) is a 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 Bernoulli-Barnes numbers. We exhibit relations among Barnes ζ-functions, Bernoulli-Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.