Divided differences and complex variations of multiple zeta-star values
Abstract
Motivated by the known derived set [1,+∞) of multiple zeta-star values, we study finite multiple star harmonic sums and their infinite-depth limits. Divided differences give a cube-integral representation of their holomorphic interpolation on Re(s)>0, which agrees with the Kawashima functions after reversal of the index. The finite-depth interpolations converge locally uniformly in s as the depth tends to infinity, uniformly over all positive integer index sequences on compact subsets. For real s>1, these limits define a bijection from the infinite positive integer indices onto (1,s]. For each admissible infinite index, they tend to the corresponding multiple zeta-star limit as Re(s)→+∞ with bounded imaginary part. For Re(s)>1, we prove strict separation of every ordered pair of finite indices and of every ordered pair of infinite indices: either the difference or the difference divided by s has positive real part throughout this half-plane, with the choice fixed by the pair. In particular, both index maps are injective. The proof uses a summation principle for functions with positive real part, established through positive integral representations and explicit kernel estimates.
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BibTeX
@misc{li2026divided,
title = {{Divided differences and complex variations of multiple zeta-star values}},
author = {Jiangtao Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.16627},
eprint = {2606.16627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16627},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Divided differences and complex variations of multiple zeta-star values
AU - Jiangtao Li
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2606.16627
UR - https://arxiv.org/abs/2606.16627
AB - Motivated by the known derived set [1,+∞) of multiple zeta-star values, we study finite multiple star harmonic sums and their infinite-depth limits. Divided differences give a cube-integral representation of their holomorphic interpolation on Re(s)>0, which agrees with the Kawashima functions after reversal of the index. The finite-depth interpolations converge locally uniformly in s as the depth tends to infinity, uniformly over all positive integer index sequences on compact subsets. For real s>1, these limits define a bijection from the infinite positive integer indices onto (1,s]. For each admissible infinite index, they tend to the corresponding multiple zeta-star limit as Re(s)→+∞ with bounded imaginary part. For Re(s)>1, we prove strict separation of every ordered pair of finite indices and of every ordered pair of infinite indices: either the difference or the difference divided by s has positive real part throughout this half-plane, with the choice fixed by the pair. In particular, both index maps are injective. The proof uses a summation principle for functions with positive real part, established through positive integral representations and explicit kernel estimates.
ER -