Some remarks about the derivative operator and generalized Stirling numbers

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.