Coincidence among sum formulas for zeta-like multiple values

TL;DR

This paper explores the relationship between two families of zeta-like multiple series, revealing a coincidence in their total sums for equal weight but complementary depths, and establishing a unified analytic-combinatorial structure.

Contribution

It introduces explicit factorial formulas and identities linking $ ho$- and $ ext{eta}$-values, demonstrating their deep combinatorial and analytic connection.

Findings

Total sums of $ ho$- and $ ext{eta}$-values coincide for equal weight and complementary depths.

Established factorial formulas and closed expressions for fixed weight and depth.

Provided integral representations and weighted sum relations for $ ext{eta}$-values.

Abstract

We study two families of zeta-like multiple series -- the multiple $\rho$-values and the multiple $\eta$-values -- defined by nested sums with shifted denominators. An explicit factorial formula for $\rho$ 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 $\rho$- and $\eta$-values coincide for equal weight but complementary depths. This correspondence provides an analytic basis for integral representations of $\eta$-values and for deriving weighted sum relations. Together, these results show that the $\rho$- and $\eta$-families form two complementary realizations of a unified analytic-combinatorial structure, bridging factorial and harmonic formulations in zeta-like multiple sums.