Higher-order tangent and secant numbers

TL;DR

This paper provides a comprehensive study of higher-order tangent and secant numbers, unifying known results with new insights, and establishing their connections to Bell polynomials and classical tangent numbers.

Contribution

It introduces new formulas and relations for higher-order tangent and secant numbers, linking them to Bell polynomials and classical tangent numbers in a simple, unified framework.

Findings

Higher-order tangent numbers form a class of partial multivariate Bell polynomials.

Secant numbers can be computed from tangent numbers.

Explicit double sum formula for higher-order tangent numbers.

Abstract

In this paper higher-order tangent numbers and higher-order secant numbers, ${\mathscr{T}(n,k)}_{n,k =0}^{\infty}$ and ${\mathscr{S}(n,k)}_{n,k =0}^{\infty}$, have been studied in detail. Several known results regarding $\mathscr{T}(n,k)$ and $\mathscr{S}(n,k)$ have been brought together along with many new results and insights and they all have been proved in a simple and unified manner. In particular, it is shown that the higher-order tangent numbers $\mathscr{T}(n,k)$ constitute a special class of the partial multivariate Bell polynomials and that $\mathscr{S}(n,k)$ can be computed from the knowledge of $\mathscr{T}(n,k)$. In addition, a simple explicit formula involving a double finite sum is deduced for the numbers $\mathscr{T}(n,k)$ and it is shown that $\mathscr{T}(n,k)$ are linear combinations of the classical tangent numbers $T_n$.