发表机构
Tarbiat Modares University（塔比巴特莫达雷斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在赫维茨级数环上引入单参数卷积，推导其性质与应用，得到相关半群生成元的显式公式，拓展了自由概率与代数结构的相关理论。
AI 中文摘要
设$R$为交换$\boldsymbol{\text{Q}}$代数，$HR$表示其赫维茨级数环。我们在$HR$上引入单参数卷积$\boldsymbol{\text{hconv}}_t$，当参数$t=-1$时其特化为内禀赫维茨乘积；正整数参数作用于有限支撑截断，在系数坐标下重现有限自由卷积。对数变换可得到加性与齐次累积量、显式逆公式、二项式、埃尔米特、拉盖尔及超几何族，以及大数定律与中心极限定理的系数类比。当$t=-1$时，独立随机变量的加法对应其$HR$中矩序列的乘法，这可应用于经典累积量、贝塔-伽马乘积及自分解律。加权系数范数将$\boldsymbol{\text{hconv}}_t$转化为交换巴拿赫代数乘积，范数连续卷积半群的生成元共轭于乘法算子，我们得到埃尔米特半群、拉盖尔半群、有限自由热生成元及莱维-辛钦生成元的显式公式。
英文摘要
Let $R$ be a commutative $\Q$-algebra and let $HR$ denote its Hurwitz series ring. We introduce a one-parameter convolution $\hconv{t}$ on $HR$ for which the specialization $t=-1$ is the intrinsic Hurwitz product. Positive integral parameters act on finite-support truncations and reproduce finite free convolution in coefficient coordinates. A logarithmic transform yields additive and homogeneous cumulants, an explicit inversion formula, binomial, Hermite, Laguerre, and hypergeometric families, and coefficientwise analogues of the law of large numbers and the central limit theorem. At $t=-1$, addition of independent random variables becomes multiplication of their moment sequences in $HR$; this gives applications to classical cumulants, beta--gamma products, and self-decomposable laws. Weighted coefficient norms turn $\hconv{t}$ into a commutative Banach algebra product. Norm-continuous convolution semigroups then have generators conjugate to multiplication operators. Explicit formulas are obtained for the Hermite and Laguerre semigroups, the finite free heat generator, and the Lévy--Khintchine generator.
