Repeated derivatives of composite functions and generalizations of the Leibniz rule

TL;DR

This paper derives closed-form formulas for repeated derivatives of functions with polynomial arguments using Hermite and Kampé de Fériet polynomials, extending Leibniz rule to products of such functions.

Contribution

It introduces new closed-form expressions for higher-order derivatives of polynomial-argument functions, generalizing Leibniz rule using special polynomials.

Findings

Closed-form formulas for derivatives of polynomial-argument functions.

Extension of Leibniz rule to products of such functions.

Practical examples demonstrating the formulas' applications.

Abstract

We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of functions of the above argument, thus giving rise to expressions which can formally be interpreted as generalizations of the familiar Leibniz rule. Finally, examples of practical interest are discussed.