Relations of multiple $t$-values of general level

TL;DR

This paper investigates the relations of multiple t-values at general levels, deriving generating functions, sum formulas, and algebraic relations, extending known results for multiple zeta and t-values.

Contribution

It introduces a unified hypergeometric function framework for multiple t-values of arbitrary level, generalizing previous results and providing new algebraic and sum formulas.

Findings

Generating functions expressed via $_3F_2$ hypergeometric functions.

Formulas for sums of multiple t-values with specific height conditions.

Symmetric sum and restricted sum formulas derived using stuffle algebra.

Abstract

We study the relations of multiple $t$-values of general level. The generating function of sums of multiple $t$-(star) values of level $N$ with fixed weight, depth and height is represented by the generalized hypergeometric function $_3F_2$, which generalizes the results for multiple zeta(-star) values and multiple $t$-(star) values. As applications, we obtain formulas for the generating functions of sums of multiple $t$-(star) values of level $N$ with height one and maximal height and a weighted sum formula for sums of multiple $t$-(star) values of level $N$ with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman's restricted sum formulas for multiple $t$-(star) values of level $N$. Some evaluations of multiple $t$-star values of level $2$ with one-two-three indices are given.