Regularized Multitangent Functions and Reduction Theorem

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.