preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202402.1143.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 19 February 2024 / Approved: 20 February 2024 / Online: 20 February 2024 (14:34:09 CET)

Euler sums are alternating (or level two) extension of multiple zeta values (MZVs). Kaneko and Tsumura initiated the study of multiple T-values (MTVs), another level two generalization, by restricting the summation indices in the definition of MZVs to a fixed parity pattern. In this paper, we shall study finite MTVs and their alternating versions which are level two and level four variations of finite MZVs, respectively. We conjecture that all finite MZVs are in the Q-span of finite MTVs which in turn apparently lie in the span of finite Euler sums, and the inclusions are both proper. We shall first provide some structural results for Euler sums of small weights, guided by the author's previous conjecture that the finite Euler sum space of weight w is isomorphic to a quotient Euler sum space of weight w. Then, by utilizing some well-known properties of the classical alternating MTVs, we shall derive a few important Q-linear relations among the finite alternating MTVs, including the reversal, linear shuffle and sum relations. We then compute the upper bound for the dimension of the Q-span of weight w finite (alternating) MTVs for w<9, both rigorously using the newly discovered relations and numerically aided by computers.

(finite) Euler sums; symmetric Euler sums; (finite) multiple T-values; symmetric multiple T-values; alternating multiple T-values

Computer Science and Mathematics, Algebra and Number Theory

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