有限与对称多重$T$-值
Finite and Symmetric Multiple $T$-Values
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中文总结 AI 辅助
受有限与对称多重zeta值的关联猜想启发，研究有限与对称多重$T$-值，证明低深度下的Hoffman型对偶关系，借助多重混合值等理论及生成函数建立相关恒等式，为两类空间的同构提供维数计算证据。
中文摘要 AI 辅助
多重$T$-值（MTV）由Kaneko与Tsumura率先研究，是具有受限乘积结构的多重zeta值（MZV）的变体。受Kaneko和Zagier提出的、关联有限MZV与对称MZV的深刻猜想（该猜想已被Zhao推广至欧拉和）的启发，我们研究了有限与对称多重$T$-值。具体而言，我们证明了有限MTV在低深度下满足Hoffman型对偶关系，证实了第二作者的若干猜想，并发现了新的恒等式族。在证明对称MTV之间的关系时，我们的工作基于Xu和Zhao的多重混合值理论、Gangl、Kaneko与Zagier发现的双zeta值关系，以及Berger等人发现的双欧拉和（加权）求和公式。随后我们利用由MTV的积分结构推导得到的生成函数，建立了有限MTV的对应关系。这些结果有助于计算由有限与对称MTV张成的$\boldsymbol{\text{Q}}$-向量空间（模掉$\boldsymbol{\text{ζ(2)}}$乘积）的维数，为两类空间之间的同构提供了有力证据。
英文摘要
The multiple $T$-values (MTVs), first studied by Kaneko and Tsumura, are a variation of the multiple zeta values (MZVs) with restricted product structure. Motivated by a deep conjecture of Kaneko and Zagier relating finite MZVs and symmetric MZVs, which was extended to Euler sums by Zhao, we study finite and symmetric multiple $T$-values. In particular, we show that finite MTVs satisfy Hoffman-type duality relations at low height, confirming several conjectures of the second author and discovering new families of identities. In proving relations among symmetric MTVs, our work builds on Xu and Zhao's theory of multiple mixed values, the relations of double zeta values discovered by Gangl, Kaneko and Zagier, and the (weighted) sum formulas of double Euler sums discovered by Berger et al. We then use generating functions derived from the integral structure of MTVs to establish the corresponding relations for finite MTVs. These results aid in the computation of dimensions of the $\Q$-vector space spanned by finite and symmetric MTVs (modulo $ζ(2)$ products), providing strong evidence for an isomorphism between the two spaces.
