Poly-Bernoulli numbers from shifted log-sine integrals

TL;DR

This paper demonstrates that shifted log-sine integrals can produce poly-Bernoulli numbers at negative integers, linking these special values to a more intrinsic zeta-type function.

Contribution

It introduces a new integral representation for poly-Bernoulli numbers using shifted log-sine integrals, expanding understanding of their intrinsic properties.

Findings

Shifted log-sine integrals' values at negative integers relate to poly-Bernoulli numbers.

Anti-diagonal sums of poly-Bernoulli numbers with negative index are obtained.

Provides a new perspective on poly-Bernoulli numbers through integral representations.

Abstract

In 1999, Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers and investigated its relation to multiple zeta values. Since the poly-Bernoulli numbers appear in this function essentially by design, it is natural to ask whether they arise as special values of more intrinsic zeta-type objects. In this article, we show that a shifted log-sine integral provides such an example. Its analytically continued values at negative integers are given by anti-diagonal sums of poly-Bernoulli numbers with negative index.