In the Frobenius problem we are given a set of coprime, positive integers a1, a2, . . . , ak, and are interested in the set of positive numbers NR that have no representation by the linear form ∑ i aixi in nonnegative integers x1, x2, . . . , xk. 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.