Higher-order tangent and secant numbers

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 $T_n$.