We consider the open unit disk \(\mathbb{D}\) equipped with the hyperbolic metric and the associated hyperbolic Laplacian \(\mathcal{L}\). For \(\lambda \in \mathbb{C}\) and \(n \in \mathbb{N}\), a \(\lambda\)-polyharmonic function of order \(n\) is a function \(f: \mathbb{D} \to \mathbb{C}\) such that \((\mathcal{L} - \lambda \, I)^n f = 0\). If \(n =1\), one gets \(\lambda\)-harmonic functions. Based on a Theorem of Helgason on the latter functions, we prove a boundary integral representation theorem for \(\lambda\)-polyharmonic functions. For this purpose, we first determine \(n^{\text{th}}\)-order \(\lambda\)-Poisson kernels. Subsequently, we introduce the \(\lambda\)-polyspherical functions and determine their asymptotics at the boundary \(\partial \mathbb{D}\), i.e., the unit circle. In particular, this proves that, for eigenvalues not in the interior of the \(L^2\)-spectrum, the zeroes of these functions do not accumulate at the boundary circle. Hence the polyspherical functions can be used to normalise the \(n^{\text{th}}\)-order Poisson kernels. By this tool, we extend to this setting several classical results of potential theory: namely, we study the boundary behaviour of \(\lambda\)-polyharmonic functions, starting with Dirichlet and Riquier type problems and then proceeding to Fatou type admissible boundary limits. In collaboration with Maura Salvatori and Massimo Picardello, JFA 2024.