Hyperderivatives of the deformation series associated with arithmetic gamma values and characteristic $p$ multiple zeta values

TL;DR

This paper establishes the algebraic independence of certain arithmetic gamma values, multiple zeta values, and their hyperderivatives in positive characteristic number theory, extending previous results with advanced algebraic tools.

Contribution

It generalizes a key result to prove algebraic independence of these special values and their hyperderivatives using modern $t$-motivic Galois theory and derivation techniques.

Findings

Proves algebraic independence of specific arithmetic gamma and multiple zeta values.

Extends previous results to include hyperderivatives of deformations.

Utilizes advanced algebraic tools like $t$-motivic Galois groups.

Abstract

In the number theory in positive characteristic, there are analogues of some special values introduced by Carlitz, Carlitz gamma values and Carlitz zeta values for instance. Each of them is further developed to arithmetic gamma values and multiple zeta values by Goss and Thakur respectively. In this paper, by generalizing a result of Chang-Papanikolas-Thakur-Yu (2010), we obtain the algebraic independence of certain arithmetic gamma values, positive characteristic multiple zeta values of restricted indices and hyperderivatives of their deformations. We prove this by using Chang-Papanikolas-Yu's derivation, Maurichat's prolongation, Namoijam's formula and Papanikolas' theory of $t$-motivic Galois group.