眾所皆知，正交多項式滿足某些遞迴關係式。在本篇論文內，我們證明其逆命題亦成立。意即若一個多項式數列 {P_{n}} 最高次數為 n 滿足下列遞迴關係式
P_{n + 1}(x) = (x - α_{n})P_{n}(x) - β_{n}P_{n - 1}(x)
其中 n 為任意非負整數，且 P_{-1}(x) = 0，P_{0}(x) = 1，則存在一些定義在實數集合上之 Borel 測度 μ 使得多項式 P_{n} 皆為對應於 μ 之正交多項式。再者，我們將討論測度之唯一性。並將針對此類多項式之正交區間，給出一些實例。
　　本篇論文可視為 M. E. H. Ismail 著作的專書 "Classical and Quantum Orthogonal Polynomials in One Variable" 第二章內容之闡述，我們的論述也參考 T. S. Chihara 的專書 "Introduction to Orthogonal Polynomials"。

It is well known that orthogonal polynomials satisfy some recurrence relations.
In this thesis, we show that the converse is also valid.
That is, if a monic polynomial sequence {P_{n}} with deg(P_{n}) = n satisfies a recurrence relation
P_{n + 1}(x) = (x - α_{n})P_{n}(x) - β_{n}P_{n - 1}(x)
for all n ∈ N ∪ {0}, and P_{-1}(x) = 0, P_{0}(x) = 1, then there exists some Borel measure μ on R such that the polynomials P_{n} are orthogonal with respect to μ.
Moreover, we discuss the uniqueness of this measure, and also the true interval of orthogonality for the polynomial sequence.
Some examples will be given.

The thesis can be viewed as an exposition of the materials in Chapter 2 of the monograph "Classical and Quantum Orthogonal Polynomials in One Variable" written by M. E. H. Ismail.
Our treatment also refers to "Introduction to Orthogonal Polynomials" written by T. S. Chihara.

1 Introduction 1

2 Polynomial sequences generated by recurrence relations 9
2.1 Orthogonal polynomial sequence (OPS) . . . . . . . . . . . . . . . . . . 9
2.2 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Quadrature formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Extension of Favard Theorem 22
3.1 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Helly's selection principle for bounded increasing functions on R . . . . 25
3.3 Extended Favard Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Continued fractions and chain sequences 35
4.1 Continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Wallis formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Chain sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 True interval of orthogonality . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Proof of Theorem 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 A lemma for symmetric measure . . . . . . . . . . . . . . . . . . . . . . 53

5 Some examples 56
5.1 Legendre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Laguerre polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.4 Chebyshev polynomials of first kind . . . . . . . . . . . . . . . . . . . . 59
5.5 Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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