  We consider the open unit disk $\mathbb{D}$ equipped with the hyperbolic
metric and the associated hyperbolic Laplacian $\mathfrak{L}$. 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 $(\mathfrak{L}-
\lambda \, I)^n f = 0$. 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.
