這篇碩士論文中, 將探討典型的正交多項式, 包含了 Chebyshev 第一型多項式, Chebyshev 第二型多項式, Laguerre 多項式以及 Jacobi 多項式.

對上述的正交多項式觀察以下性質:
(a)Rodrigues 表達式;
(b)正交基底;
(c)相對應的微分方程;
(d)遞迴關係;
(e)生成函數.
以上幾點就是為什麼這些多項式對應用數學來說是如此重要的原因.
我們的工作參考了 Folland 和 Chihara 的專著, 更進一步地對每個典型的正交多項式做更詳細的探討, 另一方面也研究了正交多項式性質的一般性.

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.

1.Introduction 1
2.Some classical orthogonal polynomials 8
2.1Chebyshev polynomials of the first kind 8
2.2Chebyshev polynomials of the second kind 11
2.3Laguerre polynomials 17
2.4Jacobi polynomials 23
3.General properties of orthogonal polynomials 30
3.1Moment function 30
3.2Recurrence relation and Favard's Theorem 37
3.3Zero of orthogonal polynomials 44
3.4Recurrence relations for Jacobi Polynomials 48
A Appendix 57
A.1 Proof of Lemma 2.16 57
A.2 Applications of orthogonal polynomials 59

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2. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

3. G. B. Folland, Fourier Analysis and Its Applications, Brook/Cole Publishing Company, California, 1992.

4. M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2009.

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