Schur type poly-Bernoulli numbers

TL;DR

This paper introduces Schur type poly-Bernoulli numbers, generalizing existing poly-Bernoulli numbers, and explores their properties, including connections to multiple zeta functions, recurrence relations, and Stirling numbers.

Contribution

It defines Schur type poly-Bernoulli numbers and investigates their properties, extending classical poly-Bernoulli numbers and linking them to multiple zeta functions.

Findings

Expression of generalized zeta functions in terms of Schur type Bernoulli numbers

Recurrence and relation formulas for Schur type Bernoulli numbers

Connections to Stirling numbers

Abstract

The poly-Bernoulli numbers and its relative are defined by the generating series using the polylogarithm series, and we call them type $B$ and $C$, respectively. As a generalization of these poly-Bernoulli numbers, we introduce Schur type poly-Bernoulli numbers and investigate their properties. First, we define a generalization of Arakawa-Kaneko multiple zeta functions and obtain their expression in terms of Schur type Bernoulli numbers. Next, under the restriction to the hook type, we define a generalization of Kaneko-Tsumura multiple zeta functions and obtain similar expression in terms of Schur type Bernoulli numbers. Lastly, we study more properties such as a recurrence formula, a relation formula between Bernoulli numbers and a description in terms of the Stirling numbers.