Multiple zeta star values on 3–2–1 indices

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 \(\zeta ^\star (\{\{2\}^m, 3, \{2\}^m, 1\}^d, \{2\}^{m+1})\), which are new and have not appeared in the literature before.