Alternating multiple \(T\)-values: weighted sums, duality, and dimension conjecture (Q6179883)

In this paper under review, the authors define some weighted sums of the alternating multiple \(T\) -values (AMTVs) by \[W_l(k,r):=\sum_{\stackrel{k_1+\cdots+k_r=k}{k_1, \ldots, k_r \geq 1}} T(k_1,\ldots, k_{l-1}, \overline{k_l},k_{l+1}, \ldots, k_{r-1}, \overline{k_r}),\] where \(0\leq l \leq r \) and \[T(k_1,\ldots, k_{r-1}, \overline{k_r}):=2^r\sum_{0<n_1<\cdots<n_r}\frac{(-1)^{n_r}}{(2n_1-1)^{k_1}\ldots (2n_r-r)^{k_r} }.\] Let \(\alpha_n=\frac{\pi^{2n}}{(2n+1)!}\). It is shown the following duality identities for any \(n,\, m,\, p \in \mathbb{N}\) and \(l \in\mathbb{N}_0\) \[\sum_{j=1}^p(-1)^j \alpha_{p- j} W_l(2 j + 2m + l - 1, 2m + l)= \sum_{j=1}^m(-1)^j \alpha_{m- j} W_l(2 j + 2p + l - 1, 2p + l)\] and \[\sum_{j=1}^p(-1)^j \alpha_{p- j} W_l(2 j + 2m + l - 3, 2m + l-1)= \sum_{j=1}^m(-1)^j \alpha_{m- j} W_l(2 j + 2p + l - 3, 2p + l-1). \] Furthermore, the authors prove a more general duality relation for general weighted sums involving AMTVs and binomial coefficients by using the method of iterated integrals. They also define an alternating convoluted \(T\) -values and an alternating version of Kaneko-Tsumura \(\psi\)-function (called Kaneko-Tsumura \(\overline{\psi}\)-function), and study some explicit relations among the alternating convoluted \(T\)-values, Kaneko-Tsumura \(\overline{\psi}\)-values and weighted sums involving AMTVs. Finally, they study the \(\mathbb{Q}\)-vector space generated by the AMTVs of any fixed weight \(w\) and provide some evidence for the conjecture that their dimensions \(\lbrace d_w\rbrace_{w\geq 1}\) form the tribonacci sequence \(1, 2, 4, 7, 13,\ldots\).