Some Symmetry and Duality Theorems on Multiple Zeta(-star) Values

In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. Leveraging Zagier’s formula for computing ${\zeta }^{\star }(\{2{\}}^p,3,\{2{\}}^q)$, we employ our theorems to establish a formula for computing ${\zeta }^{\star }(\{2{\}}^p,1,\{2{\}}^q)$ for any positive integers $p$ and $q$, along with other formulas of interest.