  Finite and Infinite-dimensional representations of symmetry algebras play a
significant role in determining the spectral properties of physical
Hamiltonians. In this paper, we introduce and apply a practical method to
construct infinite dimensional representations of certain polynomial algebras
which appear in the context of quantum superintegrable systems. Explicit
construction of these representations is a non-trivial task due to the
non-linearity of the polynomial algebras. Our method has similarities with the
induced module construction approach in the context of Lie algebras and allows
the construction of states of the superintegrable systems beyond the reach of
separation of variables. Our main focus is the representations of the
polynomial algebras underlying superintegrable systems in 2D Darboux spaces. We
are able to construct a large number of states in terms of complicated
expressions of Airy, Bessel and Whittaker functions which would be difficult to
obtain in other ways.
