Alternating multiple T-values: weighted sums, duality, and dimension conjecture

In this paper, we define some weighted sums of the alternating multiple T-values (AMTVs) and study several duality formulas for them. Then we introduce the alternating version of the convoluted T-values and Kaneko–Tsumura $$\psi $$ -function, which are proved to be closely related to the AMTVs. At the end of the paper, we 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 $$\{d_w\}_{w\ge 1}$$ form the tribonacci sequence 1, 2, 4, 7, 13, ....