连续奇数序列的$p$-数值半群
$p$-numerical semigroup of the sequence of consecutive odd integers
AI总结：
针对两类连续奇数序列，利用有界受限分拆函数的高斯多项式的对称性与单峰性，证明了Komatsu和Pandey提出的两个p-Frobenius问题猜想。
AI中文摘要：
我们针对两类连续奇数，证明了T. Komatsu与R. Pandey（发表于《Bull. Korean Math. Soc. 2025;62:1397--1409》）提出的Conjectures 7.1和7.5所对应的$p$-Frobenius问题。对于整数$r,L,n\ge0$，有界受限分拆函数$p_{\le r}^{(\le L)}(\le n)$用于计数将$n$分拆为至多$r$个部分且每个部分不超过$L$的分拆。因此，有界受限分拆函数$p_{\le 3}^{(\le a)}(\le s)$与$p_{\le 3}^{(\le a+1)}(\le s)$在证明中起核心作用，它们的生成函数是高斯多项式，其对称性与单峰性为处理这两类序列提供了通用工具。
英文摘要：
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.
