On the higher derivatives of the inverse tangent function

TL;DR

This paper derives explicit formulas and properties for higher derivatives of the inverse tangent function, exploring related polynomials and their connections to well-known sequences and polynomials.

Contribution

It introduces new explicit formulas, generating functions, and recurrence relations for polynomials derived from higher derivatives of arctan(x), linking them to classical polynomials.

Findings

Explicit formulas for higher derivatives of arctan(x)

Generating functions and recurrence relations for associated polynomials

Connections to Chebyshev, Fibonacci, Lucas, and Matching polynomials

Abstract

In this paper, we find explicit formulas for higher order derivatives of the inverse tangent function. More precisely, we study polynomials which are induced from the higher-order derivatives of arctan(x). Successively, we give generating functions, recurrence relations and some particular properties for these polynomials. Connections to Chebyshev, Fibonacci, Lucas and Matching polynomials are established.