On a new generalization of Stirling numbers and Bell polynomials

Abstract

In this paper, we introduce and study a novel class of generalized Stirling numbers depending on a real parameter and two sequences of real numbers. This class extends several known families of numbers, including the Hsu-Shiue generalized Stirling numbers $S\left(n,k;\alpha,\beta,\gamma \right)  $. We also present a combinatorial interpretation of these numbers, which allows us to derive a variety of combinatorial identities. Additionally, we establish several identities for the associated polynomials and, under suitable conditions on the defining parameters, analyze the real roots of these polynomials. Finally, we establish some properties of the differential operator acting on these polynomials.