Special functions and dual special functions in Drinfeld modules of arbitrary rank

Giacomo Hermes Ferraro

Published 2023-03-20Version 1

In a previous paper, the author showed in the context of Drinfeld-Hayes modules that the product between a special function and a Pellarin zeta function is rational, and that the latter can be interpreted as a dual special function. Both results are generalized in this paper. We use a very simple functorial point of view to interpret special functions and dual special functions in Drinfeld modules of arbitrary rank, allowing us to define a universal special function $\omega_\phi$ and a universal dual special function $\zeta_\phi$. In analogy to the Drinfeld-Hayes case, we prove that the latter can be expressed as an Eisenstein-like series over the period lattice, and that the scalar product $\omega_\phi\cdot\zeta_\phi$, an element of $\mathbb{C}_\infty\hat\otimes\Omega$, is rational. We also describe the module of special functions in the generality of Anderson modules, as already done by Gazda and Maurischat, and answer a question posed in that same paper about the invertibility of special functions in the context of Drinfeld-Hayes modules.