In the setting of a Drinfeld module \(\phi \) over a curve \(X/\mathbb {F}_q\) , we use a functorial point of view to define Anderson 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 adjoint Drinfeld module \(\phi ^*\) to define dual Anderson 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 \(X({\mathbb {C}_\infty })\setminus \{\infty \}\) to \({\mathbb {C}_\infty }\) . For all integers i, we define dot products \(\zeta _\phi \cdot \omega _\phi ^{(i)}\) as certain meromorphic differential forms over \(X_{\mathbb {C}_\infty }\setminus \{\infty \}\) 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.
