Characterizations of numerical semigroup complements via Apéry sets

In this paper, we generalize the work of Tuenter to give an identity which completely characterizes the complement of a numerical semigroup in terms of its Apéry sets. Using this result, we compute the \(m\)th power Sylvester and alternating Sylvester sums for free numerical semigroups. Explicit formulas are given for small \(m\).