TABLE OF CONTENTS The main purpose of this chapter is to introduce the classical orthogonal polynomials and provide a concise summary of their mathematical properties. These functions are solutions to certain second-order differential equations that repeatedly occur in the mathematical modeling of a wide range of dynamical systems in the natural and engineering sciences. Our interest in studying them comes from their usefulness in applications and for their ability to provide mathematical representations of a broad class of functions, i.e., they provide sets of basis functions in terms of which other functions can be expanded. In general, they satisfy second-order differential equations that belong to the class of Sturm-Liouville problems.
In the materials to follow, we state a number of relevant facts on these functions and their associated properties, but do not give the proofs. Several excellent texts already exist and it is very hard to see how their presentations can be done better. Also, our philosophy, as clearly stated in the preface, is to provide a textbook that introduces certain concepts for the main purpose of seeing how they can be applied to problems of importance in the various natural and engineering sciences; the proofs of related mathematical statements are not stressed nor actually needed for the required analysis. However, when possible, references to readily available papers and/or books are given. This is certainly the case for the topics covered in this chapter.
Three excellent references, essentially covering all the topics in this chapter, except for the last section on...
