  We introduce a symbolic representation of $r$-fold harmonic sums at negative
indices. This representation allows us to recover and extend some recent
results by Duchamp et al., such as recurrence relations and generating
functions for these sums. This approach is also applied to the study of the
family of extended Bernoulli polynomials, which appear in the computation of
harmonic sums at negative indices. It also allows us to reinterpret the Raabe
analytic continuation of the multiple zeta function as both a constant term
extension of Faulhaber's formula, and as the result of a natural
renormalization procedure for Faulhaber's formula.
