Extended Bernoulli and Stirling matrices and related combinatorial identities

TL;DR

This paper introduces new matrix representations for Bernoulli and Stirling numbers, deriving numerous combinatorial identities through Pascal-type matrices and their factorizations.

Contribution

It presents a novel matrix-based framework for Bernoulli and Stirling numbers, enabling the derivation of various identities and generalizations.

Findings

New matrix representations for Bernoulli and Stirling numbers

Derivation of numerous combinatorial identities

Factorization of Pascal-type matrices related to these numbers

Abstract

In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.