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

TL;DR

This paper introduces a conjectural $q$-analogue of classical duality for iterated integrals on the projective line minus four points, involving new $q$-integrals and a functional invariance conjecture.

Contribution

It proposes a novel $q$-duality conjecture for iterated integrals and relates it to existing dualities, proving it in specific cases.

Findings

Conjectural $q$-duality is proposed for iterated $q$-integrals.

The conjecture is related to Yamamoto's duality for $q$-polylogarithms.

The conjecture is proven in several special cases.

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.