Coincidence among sum formulas for zeta-like multiple values

TL;DR

The paper introduces two zeta-like families, the multiple $\rho$-values and the multiple $\eta$-values, defined by nested sums with shifted denominators, and proves a striking identity: the total sums by weight coincide when depths are complementary. A closed factorial formula for $\rho$-values and an integral framework linking $\eta$-sums to $\rho$-sums are developed, enabling an analytic bridge between factorial and harmonic representations. The main $\rho$–$\eta$ correspondence is established via a precise integral transformation, and several weighted sum formulas are derived, including representations of $\eta$-sums in terms of harmonic-number polynomials and finite zeta-star values. Together, these results reveal a unified analytic–combinatorial structure governing zeta-like multiple sums and provide tools for translating between the two families and their integral forms.

Abstract

We study two families of zeta-like multiple series -- the multiple $ρ$-values and the multiple $η$-values -- defined by nested sums with shifted denominators. An explicit factorial formula for $ρ$ reveals its intrinsic combinatorial structure and leads to closed expressions for fixed weight and depth. A remarkable identity emerges from a weighted-sum transformation, exhibiting a perfect discrete balance. The main theorem proves that the total sums of $ρ$- and $η$-values coincide for equal weight but complementary depths. This correspondence provides an analytic basis for integral representations of $η$-values and for deriving weighted sum relations. Together, these results show that the $ρ$- and $η$-families form two complementary realizations of a unified analytic-combinatorial structure, bridging factorial and harmonic formulations in zeta-like multiple sums.

Theorems & Definitions (14)

• Proposition 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Proposition 2.4
• Theorem 3.1
• proof
• Theorem 3.2
• proof