On a weighted sum of multiple T-values of fixed weight and depth

TL;DR

This paper investigates a weighted sum of multiple T-values, a variant of multiple zeta values, deriving its generating function in terms of depth-one T-values by solving a differential equation of Heun type.

Contribution

It provides a new explicit expression for the generating function of weighted sums of multiple T-values using differential equations of Heun type.

Findings

The generating function is expressed in terms of depth-one multiple T-values.

A differential equation of Heun type is solved to obtain this expression.

The approach links multiple T-values to special functions through differential equations.

Abstract

The multiple T-value, which is a variant of multiple zeta value of level two, is introduced by Kaneko and Tsumura. We show that the generating function of a weighted sum of the multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type.