Noncommutative factorizations of higher sine functions in positive characteristic

In this paper we describe new noncommutative factorizations of functions related to $d$-th tensor powers of Carlitz’s ${\mathbb{F}}_q[\theta ]$-module for $d\ge 1$, 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 $\mathrm{End}({\mathbb{G}}_a^d)$‘’ 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 $\mathrm{\Delta }$-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.