Various examples of exactly solvable `discrete' quantum mechanics are
explored explicitly with emphasis on shape invariance, Heisenberg operator
solutions, annihilation-creation operators, the dynamical symmetry algebras and
coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric
orthogonal polynomials satisfying difference equation versions of the
Schr\"odinger equation. Various reductions (restrictions) of the symmetry
algebra of the Askey-Wilson system are explored in detail.Comment: 46 pages, 2 figure
