Higher-order tangent and secant numbers

Abstract

In this paper, the higher-order tangent numbers and higher-order secant numbers, {I(n, k)(n,k=0)(infinity) and {I(n, k)}(n,k=0)(infinity), have been studied in detail. Several known results regarding I(n, k) and I(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 I(n, k) constitute a special class of the partial multivariate Bell polynomials and that I(n, k) can be computed from the knowledge of I(n, k). In addition, a simple explicit formula involving a double finite sum is deduced for the numbers I(n, k) and it is shown that I(n, k) are linear combinations of the classical tangent numbers T(n). (C) 2011 Elsevier Ltd. All rights reserved.