关于copartition乘积非负性猜想的进一步结果
Further results on non-negativity conjectures for copartition products
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中文总结 AI 辅助
本文针对copartition乘积的有限和无限系数非负性猜想，通过分解为局部块、证明特定情形及改进非负范围，取得了多项新进展。
中文摘要 AI 辅助
Burson和Eichhorn提出了由copartition产生的乘积的无限和有限系数非负性猜想。我们获得了若干进一步的结果。首先，这两个乘积都可以分解为基本的局部块，并且有限猜想恰好归结为其对角特化。当第一个截断参数为1或两个余数参数相等时，有限猜想得到证明。我们还证明了每个有限乘积在第三个移位符号指数以下的系数是非负的。对于固定的有限参数，系数序列被证明最终是拟多项式的；我们得到了显式的周期界、拟多项式公式的显式起始点以及每个剩余类上的首项。这产生了最终的严格正性，除了在一个已解决的对称情形中被迫的奇数次零点。对于无限猜想，我们在更广泛的条件下$a\equiv b\pmod m$证明了非负性，这包括了已知情形$a=b$。最后，对于任意的$b\mid a$，我们将均匀初始非负范围从低于$a+m$的次数改进到低于$a+2m$的次数。
英文摘要
Burson and Eichhorn proposed infinite and finite coefficientwise non-negativity conjectures for products arising from copartitions. We obtain several further results. First, both products admit decompositions into elementary local blocks, and the finite conjecture reduces exactly to its diagonal specialization. The finite conjecture is proved when the first truncation parameter is one or when the two residue parameters are equal. We also prove that every finite product has non-negative coefficients below the third shifted signed exponent. For fixed finite parameters, the coefficient sequence is shown to be eventually quasipolynomial; an explicit period bound, an explicit onset for the quasipolynomial formula, and the leading term on every residue class are obtained. This yields eventual strict positivity, apart from forced odd-degree zeros in one solved diagonal case. For the infinite conjecture, we prove non-negativity under the broader condition $a\equiv b\pmod m$, which includes the known case $a=b$. Finally, for arbitrary $b\mid a$, we improve the uniform initial non-negative range from degrees below $a+m$ to degrees below $a+2m$.
发表机构
- College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）
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