Study On Arithmetic Properties Of Recursive Sequence

The arithmetic properties of recursive sequences are hot point problems in elementary number theory and combinatorial number theory, and have important theoretical significance and application value. The well-known Fi-bonacci numbers, Lucas numbers play an important role in solving the prob-lems such as Hilbert’s10th Problem and Format’s last Theorem, and are widely used in prime decomposition, congruence, and other different areas of mathematics such as coding theory, cryptography, combinatorics, matrix theory. In1980, the famous mathematician Erdos proposed some conjec-tures involving the reciprocal sums of Fibonacci numbers. In recent years, many scholars studied the reciprocal sums of recursive sequences from dif-ferent angles. In this dissertation, the arithmetic identities of the Fibonacci polynomials, k-periodic Fibonacci numbers, higher-order linear recursive se-quence, binomial transform sequence are established by using some elemen-tary methods and techniques. These results generalize previous results in related areas, and turn out that some known results are obtained as special cases. Moreover, some conjectures proposed by Ohtsuka, Murthy, Ashbacher are completely solved. The main achievements of this dissertation are given as follows:1. By studying the Fibonacci polynomials, Lucas polynomials and their isometric sub-sequences, some round-formulas are obtained. These results are the generalizations of Fibonacci numbers and Pell numbers, which are obtained by Ohtsuka and Wenpeng Zhang.2. By using the theory of polynomial zeros distribution and error estima-tion method to study the higher-order linear recursion sequence, some identi-ties are established relating to their isometric sub-sequences and alternating series. The main purpose of this dissertation is to study the higher power of the reciprocal sums of{un}, and make a numerical analysis by MATH-EMATICA, then obtain several round-formulas relating to higher power of reciprocal sums of higher-order linear recursion sequence.3. By studying the k-period Fibonacci numbers defined by Edson and Yayenie, several identities are established relating to their isometric sub-sequences and higher power sums of reciprocal products. The study of re-ciprocal sums is extended from the linear recursive sequence to the nonlinear recursive sequence.4. By studying the recursive properties of the binomial transform se-quence, some identities of reciprocal sums of binomial transform sequence of Pell numbers are obtained. Moreover, the binomial transform sequence of generalized Tribonacci sequence is studied, the recursive relation is also es-tablished, which completely solved a series of conjectures proposed by Murthy and Ashbacher. The method of how to obtain the recurrence relation of bi-nomial transform sequence of higher-order linear recursion sequence is also given.