ABSTRACT
To understand the concept and to appreciate the utility of difference equations, we must begin with the idea of a sequence of numbers. A sequence is merely a collection of numbers that are indexed by integers. An example of a finite sequence is {5, 7, −0.3, 0.37, −9}, which is a sequence representing a collection of five numbers. By describing this sequence as being indexed by the integers, we are only saying that it is easy to identify the members of the sequence. Thus, we can identify the 2nd member in the sequence or, as another example, the 5th member. This is made even more explicit by identifying the members as y1, y2, y3, y4, and y5 respectively. Of course, sequences can be
infinite as well. Consider for example the sequence This sequence contains an infinite number of objects or elements. However, even though there are an infinite number of these objects, they are indexed by integers, and can therefore have a counting order applied to them. Thus, although we cannot say exactly how many elements there are in this sequence, we can just as easily find the 8th element or the 1093rd element.* Since we will be working with sequences, it will be useful to refer to their elements in a general way. The manner we will use to denote a sequence in general will be by using the variable y with an integer subscript. In this case the notation y1, y2, y3,…, yk,… represents a general sequence of numbers. We are often interested in discovering the values of the individual members of the sequence, and therefore must use whatever tools we have to allow us to discover their values. If the individual elements in the sequence are independent of each other, i.e., knowledge of one number (or a collection of elements) tells us nothing about the value of the element in question, it is very difficult to predict the value of the sequence element. However, many sequences are such that there exists embedded relationships between the elements; the elements are not independent, but linked together by an underlying structure. Difference equations are equations that describe the underlying structure or relationship between the sequence element; solving this family of equations means using the information about the sequence element interrelationship that is contained in the family to reveal the identity of members of the
sequence.
