On Non-central Stirling Numbers of the First Kind

TL;DR

This paper explores non-central Stirling numbers of the first kind, deriving recurrence relations, explicit formulas, and combinatorial identities, and demonstrating their role in derivatives of specific functions.

Contribution

It introduces new explicit formulas and identities for non-central Stirling numbers of the first kind, expanding understanding of their properties and applications.

Findings

Derived a recurrence relation for non-central Stirling numbers

Obtained explicit formulas for these numbers and for s(n,1,a)

Connected these numbers to derivatives of power-log functions

Abstract

It is shown in this note that non-central Stirling numbers s(n,k,a) of the first kind naturally appear in the expansion of derivatives of the product of a power function and a logarithn function. We first obtain a recurrence relation for these numbers, and then, using Leibnitz rule we obtain an explicit formula for these numbers. We also obtain an explicit formula for s(n,1,a), and then derive several combinatorial identities related to these numbers.