Functional identities of certain zeta-like functions associated to Drinfeld A-modules

In the setting of a Drinfeld module $\phi$ over a curve $X/\mathbb{F}_q$, we use a functorial point of view to define \emph{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 \textit{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 entire 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 study the case of arbitrary Anderson modules and formulate several conjectures generalizing the main theorems of this thesis.