Coincidence among sum formulas for zeta-like multiple values
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912688629612544 |
|---|---|
| author | Chen, Kwang-Wu |
| author_facet | Chen, Kwang-Wu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03431 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coincidence among sum formulas for zeta-like multiple values Chen, Kwang-Wu Number Theory 11M32, 11B83, 05A19 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. |
| title | Coincidence among sum formulas for zeta-like multiple values |
| topic | Number Theory 11M32, 11B83, 05A19 |
| url | https://arxiv.org/abs/2511.03431 |