关于Chebyshev多项式及其应用  

On Chebyshev Polynomials and Some of Their Applications

作　　者：金光滋 

机构地区：[1]日本近畿大学产业理工学部情报学科

出　　处：《商洛学院学报》2008年第5期1-17,共17页Journal of Shangluo University

摘　　要：为了展现第二类Chebyshev多项式的独特理论及其在分子轨道方面的应用,采用不完全归纳法、枚举法,研究两类Chebyshev多项式Un与Tn、正弦和余弦及其实际应用,给出了Un、Tn的三种等价定义,超几何函数表述、正交系以及在分子轨道方面的应用。研究第二类Chebyshev多项式更易于抓住问题的本质,这种处理问题的视角和论述有着深远的意义。In this paper we shall present a rather unique theory of Chebyshev polynomials of the second kind,Un,on the ground that from our point of view that it is Un that are easier to deal with.We adopt an analogy between Tn and Un and cosine and sine,which can be perceived in Exercises 3 and 5.In Section 1 we introduce Un （and Tn） in several ways：the most common way of the n-plication formula as definition; generatingfunctionology in Exercise 5,which gives a universal expression for Un involving the two-valued function √z^2-1 ;as a special case of the hypergeometric function in Exercise 8.Any one of them may be adopted as a definition and we can prove that others are equivalent to it.In Exercise 3,we find the value for 2cos 2π/5 which is the＇reciprocal of the golden ratio 1＋√5/2.The subsequent Remark 1 describes the zeroes of Un（x）, which fact is essentially applied in Section 3 to find energy levels of molecular orbitals of chainshaped polyenes; this aspect has never been elucidated in any books.In Section 2 we shall develop the elements of orthogonal systems.For the readers＇ convenience,we present many of the subsidiary results as exercises with solutions.To have a quick glance at the main stream of the theory ,they can be skipped.

关 键 词：第二类CHEBYSHEV多项式 第一类CHEBYSHEV多项式 生成函数 超几何函数 正交系 分子轨道 Hǖckel方法 

分 类 号：O156.4[理学—数学]