Abstract
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 ) and 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 ) constitute a special class of the partial multivariate Bell polynomials and that S ( n , k ) can be computed from the knowledge of T ( n , k ) . In addition, a simple explicit formula involving a double finite sum is deduced for the numbers T ( n , k ) and it is shown that T ( n , k ) are linear combinations of the classical tangent numbers T n .
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