Repeated-Index Multiple Zeta Values and Dirichlet Lambda Analogues

Our first contribution is a unified derivation of generating functions for $\zeta(\{s\}_n)$, $\zeta(\{\bar{s}\}_n)$, and $\lambda(\{s\}_n)$, obtained through Newton-type recursion formulas that refine classical symmetric-function identities. We further establish a closed combinatorial expression for the expansion coefficients of $\lambda(\{2p\}_n)$, which highlights the structural role of root-of-unity symmetries. Finally, for $\lambda(\{3\}_n)$ we prove that no such representation exists in the commutative setting, thereby motivating a non-commutative framework. Within this framework, the commutation condition $[m(x),m(-x)]=0$ leads naturally to the notion of pairwise centrality, while the sum-zero and full 2-step nilpotent constraints subsequently emerge as higher-order structural requirements ensuring global commutativity.