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arXiv · 2605.00811

Conjectural duality for iterated $q$-integrals on $\mathbb{P}^{1}$ minus four generic points

Minoru Hirose
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Abstract

We propose a conjectural $q$-analogue of the classical duality for iterated integrals on $\mathbb{P}^{1}$ 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 $q$-integrals with position-dependent $q$-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 $q$-polylogarithms. Finally, we prove the conjecture in several special cases.

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Minoru Hirose. 2026-05-01. Conjectural duality for iterated $q$-integrals on $\mathbb{P}^{1}$ minus four generic points. https://arxiv.org/abs/2605.00811

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