AI 中文总结
该研究建立stuffle正则化多重正切函数的直接解析理论，通过渐近比较等方法证明其可约化为单正切函数的有限线性组合，并推导得到正则化多重zeta值间的一族关系式。
AI 中文摘要
我们建立了stuffle正则化多重正切函数的直接解析理论，并证明它们可约化为单正切函数的有限线性组合，整个过程未使用模演算。我们首先在自然参数\textit{T}_N-\textit{H}(\textit{s})下建立了单侧截断多重Hurwitz zeta函数与其stuffle正则化之间的渐近比较关系，其中\textit{T}_N为调和截断，\textit{H}(\textit{s})为调和数函数。将该比较关系应用于对称多重正切截断，可得到亚纯的、1周期的正则化多重正切函数。\n对每个被加项进行显式部分分式分解，结合移动截断范围的渐近估计，可推导出以stuffle正则化多重zeta值表示的约化系数公式。单正切\textit{T}(1;\textit{s})的常数项与系数由\textrm{Im}\textit{s}\to\textrm{±}\textrm{∞}时的极限确定：当指标包含大于1的项时，二者均为零；而例外指标{1}^r则通过正弦商生成函数求值。作为推论，我们得到了正则化多重zeta值之间的一族关系式。
英文摘要
We develop a direct analytic theory of stuffle-regularized multitangent functions and prove their reduction to finite linear combinations of monotangent functions, without using mould calculus. We first establish an asymptotic comparison between one-sided truncated multiple Hurwitz zeta functions and their stuffle regularizations at the natural parameter \(T_N-H(s)\), where \(T_N\) is the harmonic truncation and \(H(s)\) is the harmonic-number function. Applying this comparison to symmetric multitangent truncations yields meromorphic, \(1\)-periodic regularized multitangent functions. An explicit partial-fraction decomposition of each summand, combined with asymptotic estimates for moving truncation ranges, gives formulas for the reduction coefficients in terms of stuffle-regularized multiple zeta values. The constant term and the coefficient of the monotangent \(\mathcal T(1;s)\) are determined from the limits as \(\operatorname{Im}s\to\pm\infty\): both vanish whenever the index contains an entry greater than \(1\), whereas the exceptional indices \(\{1\}^r\) are evaluated through a sine-quotient generating function. As a consequence, we obtain a family of relations among regularized multiple zeta values.
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