  In this paper, we point out connections between certain types of
indecomposable representations of $sl(2)$ and generalizations of well-known
orthogonal polynomials. Those representations take the form of infinite
dimensional chains of weight or generalised weight spaces, for which the Cartan
generator acts in a diagonal way or via Jordan blocks. The other generators of
the Lie algebras $sl(2)$ act as raising and lowering operators but are now
allowed to relate the different chains as well. In addition, we construct
generating functions, we calculate the action of the Casimir invariant and
present relations to systems of non-homogeneous second-order coupled
differential equations. We present different properties as higher-order linear
differential equations for building blocks taking the form of one variable
polynomials. We also present insight into the zeroes and recurrence relations.
