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arXiv · 2504.10100

A direct and algebraic characterization of higher-order differential operators

Włodzimierz Fechner·Eszter Gselmann
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Abstract

This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating multiple distinct operators. In contrast, we introduce a novel operator equation involving only a single $n$\textsuperscript{th}-order differential operator. We demonstrate that, under certain mild conditions, this equation serves to characterize such operators. Specifically, our results show that these higher-order differential operators can be identified as particular solutions to this single-operator identity. This approach provides a framework for understanding the algebraic structure of higher-order differential operators acting on function spaces.

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Włodzimierz Fechner, Eszter Gselmann. 2025-04-14. A direct and algebraic characterization of higher-order differential operators. https://arxiv.org/abs/2504.10100

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