Sequences and Difference Operators

Abstract

Our first experience with operators comes from infinitesimal calculus. In that setting, we are familiar with the differentiation and integration operators as they are applied to functions. As we shall see shortly, the operators of the difference calculus are applied not to functions, but to sequences of numbers. Therefore, before investigating the difference calculus, it might be useful to investigate the ways in which the sequences which underlie the difference calculus might arise. We will denote a sequence of numbers by {uk}, where the index k takes integer values. The precise indexing of a particular sequence (that is, the first and last values of k) depends upon the problem in which the sequence occurs, but is usually apparent from the problem. The notation {uk} may represent a finite set of numbers or an infinite (singly or doubly) set of numbers such as $\left\{ {uk} \right\}_{k = 0}^\infty \,or\,\,\left\{ {{w_{k,l}}} \right\}_{k,l = 0}^\infty$ . The index k here in {uk} is only a dummy variable, and we often use n, m, j,k, l, etc.