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GENERATING FUNCTIONS FOR THE QUOTIENTS OF NUMERICAL SEMIGROUPS
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 29 February 2024, pp. 1-12
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We propose generating functions,
$\textrm {RGF}_p(x)$, for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain 
$\textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use 
$\textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup 
$\langle a_1,a_2,a_3\rangle $ by p for a small positive integer p, and we characterise the generators of 
${\langle A\rangle }/{p}$ for a general numerical semigroup A and any positive integer p.
