The Decomposition Of Polynomial And Matrix Involving Second-Order Linear Recursive Sequence

This paper,using the recurrence relation and related properties of second-order linear recursive sequence (?),discussed some properties of matrices and polynomials involving (?),The main works are as follows.In the first chapter,the origin of Fibonacci and Lucas series,as well as and the definition and the ge-eral term formula of (?) were introduced.At the same time,some definitions and marks also were recommended.In the second chapter,new sequences (?)and (?)with relation to second-order linear recursive sequence(?)were defined,also some polynomials(?),(?),(?)and (?)associated with (?),(?) and (?)were defined.According to the recurrence relation and related properties of sequence(?),we studied the properties of (?) and (?),and obtained a series of factorization of (?),(?)and (?).The related results of Clark Kimberling in the Fusion,Fission,And Factors were generalized.In the third chapter,new matrices T_n and S_n related to the sequence (?) were defined.According to the diagonal sum of partitioned matrix and the relationship between the vector optimization and doubly stochastic matrix,the decomposition of matrix T_n and S_n were obtained,at the same time the eigenvalue of matrix S_n was studied.In the fourth chapter,containing the variable x matrices (?),(?)and (?)associated with the sequence (?)were defined.Using the properties of matrix (?)element,the recurrence relation and related properties associated with sequence(?),the decomposition of matrices (?),(?)and (?)were acquired.The related results of G.-Y.Lee,J.-S.kim and S.-G.Lee in The Linear Algebra of the Generalized Fibonacci Matrices were also generalized.