Stirling numbers with level $2$ and poly-Bernoulli numbers with level $2$

TL;DR

This paper introduces poly-Bernoulli numbers with level 2, explores their properties, relations, and congruences, and connects them to Stirling numbers and poly-Cauchy numbers, including a von Staudt-Clausen type theorem.

Contribution

It defines poly-Bernoulli numbers with level 2 and investigates their properties, relations, and congruences, extending classical number theory concepts.

Findings

Established properties and relations of poly-Bernoulli numbers with level 2

Connected poly-Bernoulli numbers with level 2 to poly-Cauchy numbers

Determined denominators of Bernoulli numbers with level 2 using a von Staudt-Clausen like theorem

Abstract

In this paper, we introduce poly-Bernoulli numbers with level $2$, related to the Stirling numbers of the second kind with level $2$, and study several properties of poly-Bernoulli numbers with level $2$ from their expressions, relations, and congruences. Poly-Bernoulli numbers with level $2$ have strong connections with poly-Cauchy numbers with level $2$. In a special case, we can determine the denominators of Bernoulli numbers with level $2$ by showing a von Staudt-Clausen like theorem.