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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2503.13295 |
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| _version_ | 1866913173111570432 |
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| author | Green, Nathan Pellarin, Federico |
| author_facet | Green, Nathan Pellarin, Federico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13295 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noncommutative factorizations of higher sine functions in positive characteristic Green, Nathan Pellarin, Federico Number Theory 11G09 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. |
| title | Noncommutative factorizations of higher sine functions in positive characteristic |
| topic | Number Theory 11G09 |
| url | https://arxiv.org/abs/2503.13295 |