Derivative Polynomials and Closed-Form Higher Derivative Formulae

TL;DR

This paper derives simple closed-form higher derivative formulas for eight trigonometric and hyperbolic functions using derivative polynomials and Carlitz-Scoville numbers, expanding on previous work by providing explicit formulas.

Contribution

It introduces new closed-form higher derivative formulas involving Carlitz-Scoville numbers for multiple functions, advancing symbolic differentiation techniques.

Findings

Explicit higher derivative formulas for eight functions.

Use of Carlitz-Scoville higher order tangent and secant numbers.

Extension of previous derivative polynomial methods.

Abstract

In a recent paper, Adamchik [V.S. Adamchik, On the Hurwitz function for rational arguments, Appl. Math. Comp. 187 (2007) 3--12] expressed in a closed form symbolic derivatives of four functions belonging to the class of functions whose derivatives are polynomials in terms of the same functions. In this sequel, simple closed-form higher derivative formulae which involve the Carlitz-Scoville higher order tangent and secant numbers are derived for eight trigonometric and hyperbolic functions.