A Newton Interpolation Approach to Generalized Stirling Numbers

Abstract

We employ the generalized factorials to define a Stirling-type pair $\left\{s\left(n,k;\mathbf{\alpha },\mathbf{\beta },r\right),S\left(n,k;\mathbf{\alpha },\mathbf{\beta },r\right)\right\}$ which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.