Noncommutative factorizations of higher sine functions in positive characteristic
Nathan Green, Federico Pellarin
TL;DR
This work advances the study of higher sine functions in positive characteristic by constructing new noncommutative factorizations for the $d$-th tensor powers of Carlitz’s module, denoted $\sin_A^{\otimes d}$, within algebras of Frobenius-iterated formal series. Central to the approach are motivic pairings and a novel $\Delta$-matrix calculus that formalizes higher-derivative structures; these tools yield a complete noncommutative factorization $\sin_A^{\otimes d}=\prod_{i\ge 0}^{\leftarrow}(1-\mathcal{L}_i\tau)$ and parallel finite-level identities. The results produce explicit linear relations among Carlitz polylogarithms at one and Thakur’s multiple zeta values, and they offer a motivic interpretation that mirrors integral shuffle relations in characteristic zero, adapted to the Frobenius-dynamics of function field arithmetic. Collectively, the paper links noncommutative operator factorizations, t-motive pairings, and explicit polylogarithm identities, enhancing our understanding of function-field analogues of Euler-type and shuffle relations with concrete finite-sum manifestations.
Abstract
In this paper we describe new noncommutative factorizations of functions related to $d$-th tensor powers of Carlitz's $\mathbb F_q[θ]$-module for $d\geq 1$, called higher sine functions. In recent work by the second author, factorizations of this type have been constructed for operators which are combinations of powers of a Frobenius endomorphism with coefficients ``in $\operatorname{End}(\operatorname{End}(\mathbb G_a^d))$''. In the present paper we succeed in determining factorizations with coefficients ``in $\operatorname{End}(\mathbb G_a^d)$'' which are not easily deducible from previous work. One key ingredient in obtaining this is an application of a ``motivic pairing'' that the first author introduced in recent work. Another key ingredient is the notion of ``$Δ$-matrix'' which comes into play in the analysis of the coefficients of the factorizations. Our results can be applied to explicitly describe analogues of shuffle $q^n$-powers for multiple polylogarithms at one, and to multiple zeta values of Thakur. All the identities we prove occur at the finite level.