Generating functions for the quotients of numerical semigroups

We propose a class of generating functions denoted by ${\text{RGF}}_p(x)$, which is related to the Sylvester denumerant for the quotients of numerical semigroups. Using MacMahon’s partition analysis, we can obtain ${\text{RGF}}_p(x)$ by extracting the constant term of a rational function. We use ${\text{RGF}}_p(x)$ to give a system of generators of the quotient of the numerical semigroup $⟨a_1,a_2,a_3⟩$ by $p$ for a small positive integer $p$ and we characterise the generators for $\frac{⟨A⟩}{p}$ for a general numerical semigroup $A$ and any positive integer $p$.