Higher-order tangent and secant numbers

In this paper, the higher-order tangent numbers and higher-order secant numbers, {T(n,k)}n,k=0∞ and {S(n,k)}n,k=0∞, have been studied in detail. Several known results regarding T(n,k)T(n,k) and S(n,k)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 T(n,k)T(n,k) constitute a special class of the partial multivariate Bell polynomials and that S(n,k)S(n,k) can be computed from the knowledge of T(n,k)T(n,k). In addition, a simple explicit formula involving a double finite sum is deduced for the numbers T(n,k)T(n,k) and it is shown that T(n,k)T(n,k) are linear combinations of the classical tangent numbers TnTn.