Remarkable relations between the central binomial series, Eulerian polynomials, and poly-Bernoulli numbers

TL;DR

This paper uncovers deep connections between the central binomial series at negative integers, Eulerian polynomials, and poly-Bernoulli numbers, revealing new relationships and confirming a longstanding observation.

Contribution

It establishes explicit links between central binomial series, Eulerian polynomials, and poly-Bernoulli numbers, providing new insights into their interrelations.

Findings

Expressed the central binomial series as a linear combination of polynomial values.

Identified one polynomial as a special value of the bivariate Eulerian polynomial.

Connected the other polynomial to the antidiagonal sum of poly-Bernoulli numbers.

Abstract

The central binomial series at negative integers are expressed as a linear combination of values of certain two polynomials. We show that one of the polynomials is a special value of the bivariate Eulerian polynomial and the other polynomial is related to the antidiagonal sum of poly-Bernoulli numbers. As an application, we prove Stephan's observation from 2004.