On a non-combinatorial definition of Stirling numbers

TL;DR

This paper introduces a novel, non-combinatorial approach to defining Stirling numbers using derivatives of specific functions, revealing new properties and formulas through differential calculus.

Contribution

It provides an alternative differential calculus-based definition of Stirling numbers, linking them to derivatives of functions like ln x and e^x, and derives related properties and formulas.

Findings

Derived recurrence relations for Stirling numbers

Established expansion formulas for powers and factorials

Obtained exponential generating functions for Stirling and Bell numbers

Abstract

In Combinatorics Stirling numbers may be defined in several ways. One such definition is given in [1], where an extensive consideration of Stirling numbers is presented. In this paper an alternative definition of Stirling numbers of both kind is given. Namely, Stirling numbers of the first kind appear in the closed formula for the n-th derivative of ln x. In the same way Stirling numbers of the second kind appear in the formula for the n-th derivative of f(e^x), where f(x) is an arbitrary smooth real function. This facts allow us to define Stirling numbers within the frame of differential calculus. These definitions may be interesting because arbitrary functions appear in them. Choosing suitable function we may obtain different properties of Stirling numbers by the use of derivatives only. Using simple properties of derivatives we obtain here three important properties of Stirling numbers. First are so called two terms recurrence relations, from which one can easily derive the combinatorial meaning of Stirling numbers. Next we obtain expansion formulas of powers into falling factorials, and vise versa. These expansions usually serve as the definitions of Stirling numbers, as in [1]. Finally, we obtain the exponential generating functions for Stirling and Bell numbers. As a by product the closed formulas for the $n$-th derivative of the functions f(e^x) and f(ln x) are obtained.