On some combinations of multiple zeta-star values

TL;DR

This paper proves that certain sums of multiple zeta-star values involving inserted 2's can be expressed as rational multiples of powers of pi^2, and confirms related conjectures through numerical evidence.

Contribution

It establishes explicit evaluations for sums of multiple zeta-star values with inserted 2's and verifies conjectures on their evaluations.

Findings

Sum of specific multiple zeta-star values equals rational multiples of pi^2 powers

Confirmed conjectures on multiple zeta-star value evaluations

Provided explicit formulas for sums involving inserted 2's

Abstract

We prove that the sum of multiple zeta-star values over all indices inserted two 2's into the string $(\underbrace{3,1, ..., 3,1}_{2n})$ is evaluated to a rational multiple of powers of $\pi^2$. We also establish certain conjectures on evaluations of multiple zeta-star values observed by numerical experiments.