本文選題：切比雪夫多項(xiàng)式 + 斐波那契多項(xiàng)式��； 參考：《西北大學(xué)》2015年博士論文

【摘要】：遞推序列與正交多項(xiàng)式的性質(zhì)是數(shù)論的熱門問題之一,在理論和應(yīng)用方面都有著重要的意義。著名的切比雪夫多項(xiàng)式和斐波那契多項(xiàng)式在函數(shù),逼近理論,差分方程等領(lǐng)域有著廣泛的應(yīng)用,對(duì)于密碼學(xué),組合學(xué)等數(shù)學(xué)分支以及智能傳感,衛(wèi)星定位等應(yīng)用學(xué)科的發(fā)展都有著重要的意義,再加上它們和斐波那契數(shù)列,盧卡斯數(shù)列的密切關(guān)系。因此,近些年來有有越來越多的專家學(xué)者來研究這兩類多項(xiàng)式,并且得到了許多的命題以及恒等式。但是大多數(shù)專家學(xué)者都是單獨(dú)利用一種多項(xiàng)式解決問題,研究?jī)深惗囗?xiàng)式之間的聯(lián)系的學(xué)者似乎并不多。本文結(jié)合了Falcon S以及張文鵬等專家的思想,研究了兩類多項(xiàng)式的關(guān)系,切比雪夫多項(xiàng)式的倒數(shù)無限和,切比雪夫多項(xiàng)式部分和,運(yùn)用初等方法得到了一系列包含這兩類多項(xiàng)式的恒等式,加強(qiáng)了兩類多項(xiàng)式的聯(lián)系,對(duì)國(guó)內(nèi)外專家在這一領(lǐng)域的結(jié)論進(jìn)行了延伸。本文的主要工作可以概括如下：1.應(yīng)用積分變換的方法,利用切比雪夫多項(xiàng)式以及斐波那契多項(xiàng)式的正交性對(duì)它們的關(guān)系進(jìn)行了研究,從而得到兩類多項(xiàng)式互相表示的恒等式,加強(qiáng)兩類多項(xiàng)式的聯(lián)系。同時(shí),我們利用兩類多項(xiàng)式與斐波那契數(shù)列,盧卡斯數(shù)列的關(guān)系,并運(yùn)用多項(xiàng)式的一些性質(zhì)得到一些關(guān)于斐波那契數(shù)列,盧卡斯數(shù)列的恒等式。2.利用對(duì)比系數(shù)的方法研究了關(guān)于切比雪夫多項(xiàng)式及斐波那契多項(xiàng)式的任意階導(dǎo)數(shù)與這兩類多項(xiàng)式互相表示的問題,最終得到了一些用切比雪夫多項(xiàng)式表示切比雪夫多項(xiàng)式任意階導(dǎo)數(shù)以及用斐波那契多項(xiàng)式表示斐波那契多項(xiàng)式任意階導(dǎo)數(shù)的恒等式。3.從切比雪夫多項(xiàng)式的通項(xiàng)公式及已有性質(zhì)出發(fā),運(yùn)用H. Ohtsuka處理斐波那契倒數(shù)無限和的方法,得到關(guān)于切比雪夫多項(xiàng)式的倒數(shù)無限和向下取整的一些恒等式。同時(shí)研究了切比雪夫多項(xiàng)式的部分和問題,利用兩類切比雪夫多項(xiàng)式的關(guān)系得到了一些關(guān)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.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：O156

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本文編號(hào)：1943466