Exactly solvable `discrete' quantum mechanics; shape invariance,   Heisenberg solutions, annihilation-creation operators and coherent states
  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. 
Satoru Odake and Ryu Sasaki