Some Identities For Generalized Lucas Sequence

Combinatorial identity is one of the main research topics in the field of combinatorial mathematics.Using the properties of Lucas sequences,some new identities and discussed the approximate value of series related to Lucas sequences were established.The main work of this paper can be summarized as follows:The first Chapter mainly introduced the historical background of the Fibonacci number and related concepts used in this paper.Besides,it explained some marks in this paper.In Chapter Two,according to the arctangent function and the basic properties of Fibonacci numbers and two kinds of Chebyshev polynomials,we derived some new identities for golden ratio,Fibonacci sequences,Lucas sequences as well as some identities involving the relationship between the first and second type of Chebyshev polynomials.Meanwhile,some related results of Kunle Adegoke in The golden ratio,Fibonacci numbers and BBP-type formulas were generalized.In Chapter Three,from the Binomial Theorem and the Lambert series,we derived some interesting identities involving Fibonacci numbers for reciprocal sum and generalized some related results which FengZhen Zhao put forwards in her paper Summation of certain reciprocal series related to the generalized Fibonacci and Lucas numbers.Moreover,we obtained a reciprocal sum of second-order recurrent sequences.In Chapter Four,we defined the Lucas triangles(?)and obtained some identities involvingLucas triangles by using the q-series,Cauchy q-Binomial Theorem and Rothe’s identity.Besides,we comfirmed three conjectures of Diego marques in On Some New Sums of Fibonomial Coefficients and generalized these results.