Conjectural duality for iterated $q$-integrals on $\mathbb{P}^{1}$ minus four generic points
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918477523058688 |
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| author | Hirose, Minoru |
| author_facet | Hirose, Minoru |
| contents | We propose a conjectural $q$-analogue of the classical duality for iterated integrals on $\mathbb{P}^{1}$ minus four points, arising from the involutive Möbius 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_00811 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Conjectural duality for iterated $q$-integrals on $\mathbb{P}^{1}$ minus four generic points Hirose, Minoru Number Theory 11M32 (Primary) 05A30, 11G55, 33E20 (Secondary) We propose a conjectural $q$-analogue of the classical duality for iterated integrals on $\mathbb{P}^{1}$ minus four points, arising from the involutive Möbius 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. |
| title | Conjectural duality for iterated $q$-integrals on $\mathbb{P}^{1}$ minus four generic points |
| topic | Number Theory 11M32 (Primary) 05A30, 11G55, 33E20 (Secondary) |
| url | https://arxiv.org/abs/2605.00811 |