# Repeated derivatives of tanh, sech, ... and associated polynomials

## TL;DR

This paper explores the use of special polynomials and combinatorial methods to efficiently compute repeated derivatives of functions like tanh, sech, and their hyperbolic versions, building on and extending previous research.

## Contribution

It introduces a new perspective on deriving repeated derivatives of sec, tan, and their hyperbolic forms using combinatorial analysis and special polynomials, complementing existing methods.

## Key findings

- Development of polynomial-based formulas for repeated derivatives
- Extension of combinatorial techniques to hyperbolic functions
- Review and synthesis of prior research in the field

## Abstract

Elementary problems like the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated by the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and develop a complementary point of view for the repeated derivatives of sec(.), tan(.) and for their hyperbolic counterparts.