From the text: The authors define higher order generalized Bernoulli numbers \(B_ {n, \chi}^ {(r)}\) associated to characters. The main result expresses \(B^ {(r)}_ {n, \chi}\) as an \(r\)-fold iterated integral for the measure \(\mu\) and as a sum of products of \(r\) ordinary (order 1) generalized Bernoulli numbers.