Some remarks about the derivative operator and generalized Stirling numbers

TL;DR

This paper explores the expansion of derivative operator expressions, connecting them to increasing trees and generalized Stirling numbers, and introduces new combinatorial structures called (p, k)-forests.

Contribution

It extends known derivative operator formulas to include combinatorial coefficients related to increasing trees and introduces the concept of (p, k)-forests for analyzing generalized Stirling numbers.

Findings

Derived new formulas linking derivative operators to increasing trees.

Analyzed generalized Stirling numbers and their inverses using bijective methods.

Introduced the concept of (p, k)-forests to study combinatorial properties.

Abstract

Studying expressions of the form $(f(x)D)^p$, where $D={\displaystyle \frac{d}{dx}}$ is the derivative operator, goes back to Scherk's Ph.D. thesis in 1823. We show that this can be extended as ${\displaystyle\sum \gamma_{p;a} (f^{(0)})^{a(0)+1} (f^{(1)})^{a(1)}...(f^{(p-1)})^{a(p-1)}D^{p-\sum_i i a(i)}}$}, where the summation is taken over the $p$-tuples $(a_0, a_1,..., a_{p-1})$, satisfying $\sum_{i}a(i)=p-1,\, \sum_{i}i a(i)<p$, $f^{(i)}=D^i f$ and $\gamma_{p;a}$ is the number of increasing trees on the vertex set $[0, p]$ having $a(0)+1$ leaves and having $a(i)$ vertices with $i$ children for $0<i<p$. Thus, previously known results about increasing trees, lead us to some equalities containing coefficients $\gamma_{p;a}$. In the sequel, we consider the expansion of $(x^k D)^p$ and coefficients appearing there, which are called generalized Stirling numbers by physicists. Some results about these coefficients and their inverses are discussed through bijective methods. Particularly, we introduce and use the notion of $(p, k)$-forest in these arguments.