The Fibonacci And Lucas Numbers And Orthogonal Polynomials Arithmetic Nature Of The Study

Fibonacci and Lucas numbers,firstly proposed by the famous European mathematician Leonardo Pisano in his work "Liber Abaci",are two impor-tant series and have attracted the attention of professional as well as amateur mathematicians.Although their definitions are extremely simple,they can de-duce various identities and unexpected solve relatively complex issues sometimes. Therefore they are widely applied in combinational mathematics,elementary number theory, algebra,analysis and geometric theory, especially in some as-pects of computer. In this paper,we start from the Brian Curtin's research of simplifying(?)Fn+jFm+j and (?)Ln+jLm+j,and we mainly discuss these problems by using the relations between Chebyshev polynomials with the Fi-bonacci and Lucas numbers.Furthermore,in the proving procedures,we notice that there is another orthogonal polynomial-Legendre polynomials which is as interesting as these two kinds of Chebyshev polynomials.According to its well properties,we try to establish a connection with the Fibonacci and Lucas num-bers.Thus this paper include the following aspects:1. On one hand,we generalize Brian Curtin's research of simplifying (?)Fn+jFm+j and (?)Ln+jLm+j.Firstly by using the properties of Chebyshev polynomials and combining the elementary and combinatorial method,we es-tablish identities that (?)Fn+jFm+jxj and (?)Ln+jLm+jxj.Then use the same method we simplify the formula that (?)Fn+jFm+jFe+j and (?)Ln+jLm+jFe+j. Moreover,by comparing the Brian Curtin's results and the identities we derived, we deduce a new identities involving the Fibonacci and Lucas numbers.2. On the other hand,we focus on the question that how are x2n and x2n+1 expressed by the combinatorial sums of the Legendre polynomials. This way, Chebyshev polynomials and Legendre polynomial are connected with each other. In fact, only according to the orthogonality of the Legendre polynomials, and using a combinational identity which is constantly equals to zero, we derive the definite forms of the expression skillfully. In the same time, we easily obtain a regular determinant whose elements are combination formulas.