Hyperderivatives of Periods and Quasi-Periods for Anderson 𝑡-Modules

We investigate periods, quasi-periods, logarithms, and quasi-logarithms of Anderson t t -modules, as well as their hyperderivatives. We develop a comprehensive account of how these values can be obtained through rigid analytic trivializations of abelian and A A -finite t t -modules. To do this we build on the exponentiation theorem of Anderson and investigate quasi-periodic extensions of t t -modules through Anderson generating functions. By applying these results to prolongation t t -modules of Maurischat, we integrate hyperderivatives of these values together with previous work of Brownawell and Denis in this framework.