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arXiv · 2608.14053

$p$-numerical semigroup of the sequence of consecutive odd integers

Takao Komatsu·Sungjin Hyun·Kyunghwan Song
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Abstract

We prove the $p$-Frobenius problems proposed as Conjectures 7.1 and 7.5 developed by T. Komatsu and R. Pandey (Bull. Korean Math. Soc. 2025;62:1397--1409.) for two families of consecutive odd integers. For integers $r,L,n\ge0$, the bounded restricted partition function $p_{\le r}^{(\le L)}(\le n)$ counts partitions of $n$ into at most $r$ parts, each at most $L$. Thus the bounded restricted partition functions $p_{\le 3}^{(\le a)}(\le s)$and $p_{\le 3}^{(\le a+1)}(\le s)$ play central roles in the proofs. Their generating functions are Gaussian polynomials, whose symmetry and unimodality provide a common tool for treating both families.

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Takao Komatsu, Sungjin Hyun, Kyunghwan Song. 2026-08-14. $p$-numerical semigroup of the sequence of consecutive odd integers. https://arxiv.org/abs/2608.14053

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