Three simple reduction formulas for the denumerant functions

Let A be a nonempty set of positive integers. The restricted partition function \(p_A(n)\) $p_A(n)$ denotes the number of partitions of n with parts in A. When the elements in A are pairwise relatively prime positive integers, Ehrhart, Sertöz–Özlük, and Brown–Chou–Shiue derived three reduction formulas for \(p_A(n)\) $p_A(n)$ for A with three parameters. We extend their findings for general A using the Bernoulli–Barnes polynomials.