Summary: A method is given whereby special functions of hypergeometric type and their generalizations can be derived in a systematic fashion from Lie algebras of difference and differential operators in one complex variable. The method automatically yields contour integral representations, generating functions, recurrence formulas, integral identities, and infinite sum identities for each function treated. This approach unites Vilenkin's integral transform method for special functions and the older factorization method into a single flexible tool.