  We present a symbolic representation for the poly-Bernoulli numbers. This
allows us to prove several new iterated integral representations for the
poly-Bernoulli numbers, including an integral transform of the Bernoulli-Barnes
numbers. We also deduce some new recurrences for the poly-Bernoulli numbers.
Finally, we use these results to present a new iterated integral representation
for the Arakawa-Kaneko zeta function, including a nonlinear integral transform
of the Barnes zeta function.
