A generalization of Stirling numbers

TL;DR

This paper extends Stirling numbers of the first kind to arbitrary real numbers, revealing new combinatorial properties and analogous behaviors for negative integer cases, broadening understanding of these classical numbers.

Contribution

It introduces a generalization of Stirling numbers to real numbers, uncovering new combinatorial properties and extending known properties to negative integers.

Findings

New combinatorial properties for classical Stirling numbers

Analogous properties for Stirling numbers with negative integers

Broader mathematical framework for Stirling numbers

Abstract

We generalize the Stirling numbers of the first kind $s(a,k)$ to the case where $a$ may be an arbitrary real number. In particular, we study the case in which $a$ is an integer. There, we discover new combinatorial properties held by the classical Stirling numbers, and analogous properties held by the Stirling numbers $s(n,k)$ with $n$ a negative integer. On g\'{e}n\'{e}ralise ici les nombres de Stirling du premier ordre $s(a,k)$ au cas o\`u $a$ est un r\'eel quelconque. On s'interesse en particulier au cas o\`u $a$ est entier. Ceci permet de mettre en evidence de nouvelles propri\'et\'es combinatoires aux quelles obeissent les nombres de Stirling usuels et des propri\'et\'es analougues auquelles obeissent les nombres de Stirling $s(n,k)$ o\`u $n$ est un entier n\`egatif.