Lie theory and difference equations. I

A factorization method is constructed for sequences of second-order linear difference equations in analogy with the factorization method for differential equations. Six factorization types are established and recursion relations are obtained for various classes of special functions, among which are the hypergeometric functions and their limits, and the classical polynomials of a discrete variable: Tchebycheff, Krawtchouk, Charlier, Meixner, and Hahn. It is shown that the factorization method is a disguised form of Lie algebra representation theory.