Orthogonalpolynomialsandsomeoftheirmodern Developments

The theory of orthogonal polynomials is a very active research field in the world. It is widelyapplied in mathematics, physics, and other branches of science and technology. Many break-throughs in mathematics, such as De Branges'proof of Bieberbach conjecture, have used someimportant results from orthogonal polynomials. In recent years, orthogonal polynomials havemade rapid progress under the leadership of mathematicians like Richard Askey, George Andrews,Mourad Ismail etc. The most noticeable achievement is the discovery of Askey-Wilson orthogonalpolynomials. These polynomials are the most general orthogonal polynomials explicitly knowntoday. Maple with its powerful symbolic and numerical capability and its impressive graphics,has become one of the three most widely used mathematical software in the world (other two areMathematica and Matlab). Maple graphics and animations can provide intuitions to some veryabstract properties of orthogonal polynomials. Based on the celebrated WZ algorithm, Maple canalso prove vast number of identities. In this work we use Maple to demonstrate some of the com-mon properties of the classical orthogonal polynomials and prove some identities related to thesepolynomials. Askey-Wilson orthogonal polynomials is the instrumental to many important workin the current mathematical research. Considering the fact that this thesis is already long, however,we limit ourselves to only introduce the basic properties of these polynomials.This thesis is organized into four chapters. In the first chapter we will list some of the neces-sary background material such as Euler's Gamma-function, Beta-functions, hypergeometric series,and some of the basic properties for general orthogonal polynomials, etc.. We focus ourselves onJacobi polynomials and their special cases with Maple in the second chapter. While in the thirdchapter, taking advantage of Maple, we concentrate on Laguerre polynomials, Hermite polynomi-als and the expansions of orthogonal polynomials. In the last chapter we introduce Askey-Wilsonpolynomials.