Carlitz operators and higher polylogarithm identities

We study a higher dimension generalization of Carlitz's polynomials, first introduced by Papanikolas, and compute an (Formula presented.) -adic limit of a sequence of normalizations, relating it to the exponential function of an Anderson module that we completely describe. We further investigate factorization properties of exponential functions generalizing recent results in the dimension one case; see [Chung, Ngo Dac, and Pellarin, Adv. Math. 422 (2023); Pellarin, Arithmetic and geometry over local fields, Springer, Berlin, 2020]. The factorizations we consider take place in certain non-commutative algebras of operators and are, up to our knowledge, the first examples of factorizations related to exponential functions associated with higher dimension Anderson modules. We explain, as an application, how to deduce explicit and numerically computable non-trivial linear dependence relations connecting polylogarithms associated to the Carlitz's module, and powers of the Carlitz fundamental period. While the existence of such identities is known and certainly central in this theory, the way they can be derived from factorization properties of higher dimension generalization of Carlitz's polynomials is completely novel, and the results are, in the opinion of the author, surprisingly simple and explicit.