AbstractsMathematics

We shall study the mathematical properties of some classical orthogonal polynomials: Chebyshev polynomials of the first kind, Chebyshev polynomials of the second kind, Laguerre polynomials and Jacobi polynomials. All of these orthogonal polynomials observe the properties: (a)Rodrigues formula; (b)Orthogonal basis; (c)Associated differential equations; (d)Recurrence relations; (e)generating functions. These properties are the main reasons why these polynomials are important in mathematics and in applications. We shall study these properties for each of the above classical orthogonal polynomials in detail. On the other hand, we shall study the general relationship among these properties. Our work mainly follows the monographs of Folland and Chihara. Advisors/Committee Members: Chun-Kong Law (committee member), Kung-Chien Wu (chair), Hsin-Yuan Huang (chair).