Some remarks about the derivation operator and generalized Stirling numbers

Details Some remarks about the derivation operator and generalized Stirling numbers Some remarks about the derivation operator and generalized Stirling numbers

2010-12-17

arxiv.org/abs/1012.3948v1

Mathematics

Combinatorics

16 pages, 2 figures

Scientific paper

Studying expressions of the form $(f(x)D)^p$, where $D={\displaystyle \frac{d}{dx}}$ is the derivation 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)

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