Recurrences for values of the Hurwitz type poly-Bernoulli numbers and polynomials

TL;DR

This paper introduces a new class of generalized Hurwitz type poly-Bernoulli numbers and polynomials, deriving algorithms and combinatorial formulas, and linking them to generalized Stirling numbers and graph theory.

Contribution

It presents a novel generalization of Hurwitz type poly-Bernoulli numbers and polynomials, along with algorithms and combinatorial formulas for their evaluation.

Findings

Derived algorithms for evaluating Hurwitz type poly-Bernoulli numbers.

Established combinatorial formulas for these numbers and polynomials.

Connected generalized Stirling numbers to graph theory.

Abstract

The main object of this paper is to investigate a new class of the generalized Hurwitz type poly-Bernoulli numbers and polynomials from which we derive some algorithms for evaluating the Hurwitz type poly-Bernoulli numbers and polynomials. By introducing a new generalization of the Stirling numbers of the second kind, we succeed to establish some combinatorial formulas for the generalized Hurwitz type poly-Bernoulli numbers and polynomials with negative upper indices. Moreover, we give a connection between the generalized Stirling numbers of the second kind and graph theory.