Recurrences for the derivative polynomials for tangent and secant

TL;DR

This paper explores the relationships and expansions between derivative polynomials for tangent and secant functions, revealing connections with Eulerian and Chebyshev polynomials through basis set analysis.

Contribution

It introduces new expansions of tangent derivative polynomials in terms of secant derivatives and explores their links with Eulerian and Chebyshev polynomials.

Findings

Expansion of tangent derivative polynomials in secant basis

Relationships between alternating derivative and Eulerian polynomials

Shared properties of alternating derivative polynomials and Chebyshev polynomials

Abstract

In this paper, we choose the derivative polynomials for tangent and secant as basis sets of polynomial space. From this viewpoint, we first give an expansion of the derivative polynomials for tangent in terms of the derivative polynomials for secant, and we then present a result in the reverse direction. We also discuss the relationships between alternating derivative polynomials and Eulerian polynomials. As applications, we give certain expansions of the alternating derivative polynomials, which indicate that the alternating derivative polynomials share more properties with the Chebyshev polynomials.