关于导数算子和广义Stirling数的几点注记
Some remarks about the derivative operator and generalized Stirling numbers
摘要：
研究$(f(x)D)^p$形式的表达式，其中$D={\displaystyle \frac{d}{dx}}$是导数算子，可以追溯到Scherk的博士学位。1823年的论文。我们表明，这可以被扩展为${\displaystyle\sum γ_{p;a} (f^{(0)})^{a(0)+1} (f^{(1)})^{a(1)}...(f^{(p-1)})^{a(p-1)}D^{p-\sum_i i a(i)}}$ }，其中的总和是在$p$ -元组$(a_0, a_1,..., a_{p-1})$，满足$\sum_{i}a(i)=p-1,\, \sum_{i}i a(i)<p$，$f^{(i)}=D^i f$和$γ_{p;a}$是增加树的顶点集$[0, p]$有$a(0)+1$叶和$a(i)$顶点与$i$的$0<i<p$的孩子。因此，以前已知的结果增加树，导致我们的一些等式包含系数$γ_{p;a}$。在第二章中，我们考虑了$(x^k D)^p$的展开式以及其中出现的系数，物理学家称之为广义Stirling数。利用双射方法讨论了这些系数及其逆的一些结果。特别地，我们在这些论证中引入并使用了$(p, k)$ -森林的概念。
英文摘要：
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 ${\displaystyle\sum γ_{p;a} (f^{(0)})^{a(0)+1} (f^{(1)})^{a(1)}...(f^{(p-1)})^{a(p-1)}D^{p-\sum_i i a(i)}}$}, 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 $γ_{p;a}$ 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 $γ_{p;a}$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
