Conjectural duality for iterated $q$-integrals on $\mathbb{P}^{1}$ minus four generic points
Number Theory
2026-05-04 v1
Abstract
We propose a conjectural -analogue of the classical duality for iterated integrals on minus four points, arising from the involutive M\"{o}bius transformation which exchanges the four marked points in pairs. To this end, we introduce iterated -integrals with position-dependent -shifts of the parameters and define a functional on admissible words in the six pairwise letters. The conjecture states that this functional is invariant under a natural anti-automorphism of the word algebra. We relate the conjecture to Yamamoto's duality for one-variable multiple -polylogarithms. Finally, we prove the conjecture in several special cases.
Cite
@article{arxiv.2605.00811,
title = {Conjectural duality for iterated $q$-integrals on $\mathbb{P}^{1}$ minus four generic points},
author = {Minoru Hirose},
journal= {arXiv preprint arXiv:2605.00811},
year = {2026}
}