A characterization of the Frobenius problem and its application to arithmetic progressions

In the Frobenius problem we are given a set of coprime, positive integers $a_1, a_2,...,a_k$, and are interested in the set of positive numbers NR that have no representation by the linear form $\sum_i a_ix_i$ in nonnegative integers $x_1, x_2,...,x_k$. We give a functional relationship that completely characterizes the set NR, and apply it to the case when the numbers are in an arithmetic progression.