  We develop our method to prove quantum superintegrability of an integrable 2D
system, based on recurrence relations obeyed by the eigenfunctions of the
system with respect to separable coordinates. We show that the method provides
rigorous proofs of superintegrability and explicit constructions of higher
order generators for the symmetry algebra. We apply the method to 5 families of
systems, each depending on a parameter $k$, including most notably the caged
anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a
deformed Kepler-Coulomb system, and we give proofs of quantum
superintegrability for all rational values of $k$, new for 4 of these systems.
In addition, we show that the explicit information supplied by the special
function recurrence relations allows us to prove, for the first time in 4
cases, that the symmetry algebra generated by our lowest order symmetries
closes and to determine the associated structure equations of the algebras for
each $k$. We have no proof that our generating symmetries are of lowest
possible order, but we have no counterexamples, and we are confident we can can
always find any missing generators from our raising and lowering operator
recurrences. We also get for free, one variable models of the action of the
symmetry algebra in terms of difference operators. We describe how the
St\"ackel transform acts and show that it preserves the structure equations.
