关于斐波那契多项式与切比雪夫多项式的一些恒等式

发布时间：2018-05-27 19:28

本文选题：切比雪夫多项式 + 斐波那契多项式　；参考：《西北大学》2015年博士论文

【摘要】：递推序列与正交多项式的性质是数论的热门问题之一,在理论和应用方面都有着重要的意义。著名的切比雪夫多项式和斐波那契多项式在函数,逼近理论,差分方程等领域有着广泛的应用,对于密码学,组合学等数学分支以及智能传感,卫星定位等应用学科的发展都有着重要的意义,再加上它们和斐波那契数列,卢卡斯数列的密切关系。因此,近些年来有有越来越多的专家学者来研究这两类多项式,并且得到了许多的命题以及恒等式。但是大多数专家学者都是单独利用一种多项式解决问题,研究两类多项式之间的联系的学者似乎并不多。本文结合了Falcon S以及张文鹏等专家的思想,研究了两类多项式的关系,切比雪夫多项式的倒数无限和,切比雪夫多项式部分和,运用初等方法得到了一系列包含这两类多项式的恒等式,加强了两类多项式的联系,对国内外专家在这一领域的结论进行了延伸。本文的主要工作可以概括如下：1.应用积分变换的方法,利用切比雪夫多项式以及斐波那契多项式的正交性对它们的关系进行了研究,从而得到两类多项式互相表示的恒等式,加强两类多项式的联系。同时,我们利用两类多项式与斐波那契数列,卢卡斯数列的关系,并运用多项式的一些性质得到一些关于斐波那契数列,卢卡斯数列的恒等式。2.利用对比系数的方法研究了关于切比雪夫多项式及斐波那契多项式的任意阶导数与这两类多项式互相表示的问题,最终得到了一些用切比雪夫多项式表示切比雪夫多项式任意阶导数以及用斐波那契多项式表示斐波那契多项式任意阶导数的恒等式。3.从切比雪夫多项式的通项公式及已有性质出发,运用H. Ohtsuka处理斐波那契倒数无限和的方法,得到关于切比雪夫多项式的倒数无限和向下取整的一些恒等式。同时研究了切比雪夫多项式的部分和问题,利用两类切比雪夫多项式的关系得到了一些关于其部分和的公式。
[Abstract]:The properties of recursive sequences and orthogonal polynomials are one of the most popular problems in number theory and have great significance in both theory and application. The famous Chebyshev polynomials and Fibonacci polynomials are widely used in the fields of function, approximation theory, difference equation and so on. The development of satellite positioning and other applied disciplines is of great significance, in addition to their close relationship with Fibonacci sequence and Lucas sequence. Therefore, in recent years, more and more experts and scholars have studied these two kinds of polynomials, and got many propositions and identities. However, most experts and scholars use one polynomial alone to solve the problem, and few scholars study the relationship between the two kinds of polynomials. In this paper, combining the ideas of Falcon S and Zhang Wenpeng, we study the relations between two kinds of polynomials, the reciprocal infinite sum of Chebyshev polynomials, the partial sum of Chebyshev polynomials. A series of identities containing these two kinds of polynomials are obtained by using the elementary method. The relations between the two kinds of polynomials are strengthened and the conclusions of domestic and foreign experts in this field are extended. The main work of this paper can be summarized as follows: 1. By using the method of integral transformation and using the orthogonality of Chebyshev polynomials and Fibonacci polynomials, the identities of the two kinds of polynomials are obtained and the relations between the two kinds of polynomials are strengthened. At the same time, we use the relations between two kinds of polynomials and Fibonacci sequence, Lucas sequence, and obtain some identities about Fibonacci sequence and Lucas sequence by using some properties of polynomial. By using the method of contrast coefficient, the problem of the representation of any order derivative of Chebyshev polynomial and Fibonacci polynomial and these two kinds of polynomials is studied. Finally, we obtain some identities of Chebyshev polynomials representing any order derivatives of Chebyshev polynomials and Fibonacci polynomials representing arbitrary derivatives of Fibonacci polynomials. Based on the general formula of Chebyshev polynomials and the existing properties, by using the method of H. Ohtsuka to deal with the infinite sum of the inverse of Fibonacci, some identities about the inverse infinity and downward integral of Chebyshev polynomials are obtained. At the same time, the partial sum of Chebyshev polynomials is studied, and some formulas about the partial sum of Chebyshev polynomials are obtained by using the relations between two kinds of Chebyshev polynomials.
【学位授予单位】：西北大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O156

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