Applications of the cosecant and related numbers

Abstract

Power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a power series expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c k, converge rapidly to zero as k→∞. It is then shown how recent advances in obtaining meaningful values from divergent series can be modified to determine exact nume..