Multiple zeta star values on 3-2-1 indices

TL;DR

This paper introduces a new, simplified method for evaluating multiple zeta star values on 3-2-1 indices, improving computational efficiency and providing explicit formulas using generating functions and Bell polynomials.

Contribution

It offers a novel derivation approach for known formulas and computes explicit evaluations of specific multiple zeta star values with two different methods.

Findings

Derived simpler formulas for multiple zeta star values on 3-2-1 indices.

Provided explicit evaluations of specific multiple zeta star values.

Introduced two methods: generating functions and Bell polynomials, for computations.

Abstract

In 2008, Muneta found explicit evaluation of the multiple zeta star value $\zeta^\star(\{3, 1\}^d)$, and in 2013, Yamamoto proved a sum formula for multiple zeta star values on 3-2-1 indices. In this paper, we provide another way of deriving the formulas mentioned above. It is based on our previous work on generating functions for multiple zeta star values and also on constructions of generating functions for restricted sums of alternating Euler sums. As a result, the formulas obtained are simpler and computationally more effective than the known ones. Moreover, we give explicit evaluations of $\zeta^\star(\{\{2\}^m, 3, \{2\}^m, 1\bigr\}^d)$ and $\zeta^\star(\{\{2\}^m, 3, \{2\}^m, 1\}^d, \{2\}^{m+1})$ in two ways. The first is based on computation of product of generating functions, while the second uses properties of Bell polynomials.