A duality result about special functions in Drinfeld modules of arbitrary rank

In the setting of a Drinfeld module $\phi$ over a curve $X\mathrm{/}{\mathbb{F}}_q$, we use a functorial point of view to define $\mathit{\text{Anderson}}\mathit{\text{eigenvectors}}$, a generalization of the so called “special functions” introduced by Anglès, Ngo Dac and Tavares Ribeiro, and prove the existence of a universal object ${\omega }_{\phi }$. We adopt an analogous approach with the dual Drinfeld module ${\phi }^{\ast }$ to define $\mathit{\text{dual}}\mathit{\text{Anderson}}\mathit{\text{eigenvectors}}$. The universal object of this functor, denoted by ${\zeta }_{\phi }$, is a generalization of Pellarin zeta functions, can be expressed as an Eisenstein-like series over the period lattice, and its coordinates are analytic functions from $X({\mathbb{C}}_{\mathrm{\infty }})\setminus \mathrm{\infty }$ to ${\mathbb{C}}_{\mathrm{\infty }}$. For all integers $i$ we define dot products ${\zeta }_{\phi }\cdot {\omega }_{\phi }^{(i)}$ as certain meromorphic differential forms over $X_{{\mathbb{C}}_{\mathrm{\infty }}}\setminus \mathrm{\infty }$, and prove they are actually rational. This amounts to a generalization of Pellarin’s identity for the Carlitz module, and is linked to the pairing of the $A$-motive and the dual $A$-motive defined by Hartl and Juschka. Finally, we develop an algorithm to compute the forms ${\zeta }_{\phi }\cdot {\omega }_{\phi }^{(i)}$ when $X={\mathbb{P}}^1$, and prove a conjecture of Gazda and Maurischat about the invertibility of special functions for Drinfeld modules of rank $1$.