# A symbolic approach to the poly-Bernoulli numbers

## TL;DR

This paper introduces a symbolic framework for poly-Bernoulli numbers, deriving new integral representations, recurrences, and connections to zeta functions, advancing theoretical understanding in number theory.

## Contribution

It provides a novel symbolic approach that yields new integral formulas and recurrences for poly-Bernoulli numbers and related zeta functions.

## Key findings

- New iterated integral representations for poly-Bernoulli numbers
- Integral transform of Bernoulli-Barnes numbers
- Recurrences for poly-Bernoulli numbers

## Abstract

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.