ABSTRACT
Interest in difference equations goes a long way back to the times before the discovery of differential and integral calculus. For instance, the famous sequence 0,1,1,2,3,5,8, … that appeared in the work of Fibonacci (ca. 1202) is the solution of a difference equation, namely xn = xn−1 + xn−2 with given initial values x−1 = 0, x0 = 1. This difference equation also generates the well-known Lucas numbers if x−1 = 1, x0 = 3. Even after the invention of the concept of derivative until around the mid-twentieth century, difference equations found numerous applications in numerical analysis where they were used in the solution of algebraic and differential equations. Indeed, the celebrated Newton’s method for finding roots of scalar equations is an example of a difference equation, as is the equally famous Euler’s method for estimating solutions of differential equations through estimation of the derivative by a finite difference (Burden and Fairs, 1997). These are just two among many other and more refined difference methods for dealing with complex problems in calculus and differential equations. By the mid-twentieth century, the theory of linear difference equations had been developed in sufficient detail to rival, indeed parallel, its differential analog. This theory had already been put to use in the 1930s and 1940s by economists (Hicks, 1965; Samuelson, 1939) in their analyses of discrete-time models of the business cycle.
