Some Identities Of Fibonacci Polynomials And Chebyshev Polynomials

The properties of orthogonal polynomials and recursive sequences are popu-lar in number theory. They are important in theoretical research and application. The famous Chebyshev polynomials and Fibonacci polynomials are widely used in the field of function, approximation theory and difference equation. They also promote the development of both the branch of mathematics such as cryptogra-phy, combinatorics and application of discipline such as intelligent sensing and satellite positioning. Furthermore, they are close to the Fibonacci numbers and Lucas numbers. Therefore, many authors have investigated them and get many properties and identities. However, most of the experts and scholars use only one kind of the polynomials to solve problems. There are only a few experts who study the relationships between these polynomials. In this dissertation, we combine some experts’ ideas such as Sergio Falcon and Wenpeng Zhang, to study the relationships between these polynomials, the reciprocal sums of Chebyshev polynomials, the partial sum of Chebyshev polynomials. We use the elementary method to get a lot of properties involving these polynomials. These results strengthen the connections of two kinds of polynomials and generalize previous results in related areas. The main achievements of this dissertation are given as follows:1.By using the way of integral transformation, we study the relationship of the Chebyshev polynomials and Fibonacci polynomials, according to their orthogonality. In order to strengthen the connections of two kinds of polynomials, we get some identities of the first and second kind Chebyshev polynomials in terms of the Fibonacci polynomials and vice versa. At the same time, we get some identities involving the Fibonacci numbers and the Lucas numbers by using the connection of Chebyshev polynomials, Fibonacci polynomials and these series.2.By using the way of undetermined coefficient, we study the problems of the rth derivative of Fibonacci polynomials and Chebyshev polynomials being represented by these polynomials. Finally, we get some identities about the rth derivative of Fibonacci polynomials and the first kind of Chebyshev polynomials being represented by these polynomials.3.By using the way which H. Ohtsuka solved the problem of the reciprocal sums of Fibonacci numbers, we get some identities about the reciprocal sums of Chebyshev polynomials. At the same time, we study the partial sum of Cheby-shev polynomials. Then we get some identities about the partial sum of Cheby-shev polynomials by the relation of two kinds of Chebyshev polynomials.