Relations between the powers of some special matrices and generalized fibonacci and lucas sequences

This thesis consists of six chapters. In the first chapter, some concepts and studies in the literature related to the subject of the thesis are mentioned, and then main purpose of the thesis is explained. In the second chapter, definitions and theorems that will be used in the continuation of the study on matrices and number sequences are given. The third chapter is the first part of the original part of the thesis. This chapter consists of three sections. In the first section, the relationship between the generalized Fibonacci sequence and 3 × 3 dimensional matrices with eigenvalues 𝛼, 𝛽 and 𝑟 is examined. Here 𝑝 and 𝑞 are nonzero real numbers that satisfy the condition 𝑝 2 + 4𝑞 > 0, 𝛼 and 𝛽 are the roots of the equation 𝑥^2 − 𝑝𝑥 − 𝑞 = 0, and 𝑟 is a nonzero real number with 𝑟^2 − 𝑝𝑟 − 𝑞 ≠ 0. First, the system of equations required to obtain such matrices is shown. Then, with the help of matrix diagonalization, the relation between the powers of such matrices and generalized Fibonacci numbers is obtained. Here, the purpose is to obtain matrices whose terms and power relations would be simpler for usability. It is studied to obtain such matrices by choosing appropriate eigenvectors. Additionally, a special matrix and power relation of the desired type is obtained. In the second section, similar to the previous section, the relationship between matrices and the generalized Fibonacci sequence is examined. Therefore, matrices satisfying the equation (𝑥 2 − 𝑝𝑥 − 𝑞)(𝑥 − 𝑟) = 0, or equivalently, the equation 𝑥^3 − (𝑟 + 𝑝)𝑥^2 + (𝑟𝑝 − 𝑞)𝑥 + 𝑞𝑟 = 0 are studied, where 𝑝, 𝑞 and 𝑟 are nonzero real numbers with 𝑝^2 + 4𝑞 > 0 and 𝑟^2 − 𝑝𝑟 − 𝑞 ≠ 0. First, the theorem showing that the integer powers of all matrices 𝑋 satisfying the condition 𝑋^3 − (𝑟 + 𝑝)𝑋^2 + (𝑟𝑝 − 𝑞)𝑋 + 𝑞𝑟𝐼 = 𝟎 can be found with the help of the generalized Fibonacci sequence is obtained. Then this theorem is used for some suitable 3 × 3 dimensional commutative matrices. Thanks to these matrices obtained, some identities are obtained for the terms of the generalized Fibonacci sequence. Additionally, the relationships between generalized Fibonacci sequences established with different indices are examined using the obtained matrices. Some identities are obtained between the terms of the sequence {𝑈𝑛(𝑝, 𝑞)}, which was established with the appropriate real numbers 𝑝, 𝑞 and 𝑟, and the sequence {𝑈𝑛(𝑝1, 𝑞1)}, which was established with 𝑝1 = −𝑞 + 𝑝𝑟 – 𝑟^2 and 𝑞1 = 𝑞𝑟(𝑝 − 𝑟). In the last section, the relations between the generalized Fibonacci and Tribonacci sequences are examined. First a special generalized Tribonacci sequence {𝑇𝑛(𝑝, 𝑞, 𝑟)} is defined. Then, it is shown that the powers of matrices 𝑋 satisfying the condition 𝑋^3 − (𝑟 + 𝑝)𝑋^2 + (𝑟𝑝 − 𝑞)𝑋 + 𝑞𝑟𝐼 = 𝟎 can be obtained with the help of the sequence {𝑇𝑛(𝑝, 𝑞, 𝑟)}. Considering this result and the result obtained for the generalized Fibonacci sequence in the previous section, it is obtained that the powers of the matrices 𝑋 satisfying the condition 𝑋^3 − (𝑟 + 𝑝)𝑋^2 + (𝑟𝑝 − 𝑞)𝑋 + 𝑞𝑟𝐼 = 𝟎 can be found with the help of both the generalized Fibonacci sequence {𝑈𝑛(𝑝, 𝑞)} and the generalized Tribonacci sequence {𝑇𝑛(𝑝, 𝑞, 𝑟)}. Based on this, some relations between the terms of the sequence {𝑈𝑛(𝑝, 𝑞)} and the sequence {𝑇𝑛(𝑝, 𝑞, 𝑟)} are obtained. Finally, an application is made with the help of the obtained results. It is shown that a special solution of the equation 𝑥𝑦 + 𝑥 + 𝑦 = 10^𝑛 − 1 for all integers 𝑛 can be obtained with the help of the sequence {𝑇𝑛(𝑝, 𝑞, 𝑟)}. In the fourth chapter, the relations between generalized Fibonacci and Lucas sequences and matrices are studied. In the first section, a new property has been obtained for matrices 𝑋 satisfying the condition 𝑋^2 − 𝑝𝑋 − 𝑞𝐼 = 𝟎, where 𝑝 and 𝑞 are nonzero real numbers with 𝑝^2 + 4𝑞 > 0. A result is obtained for the linear combination of integer powers of such matrices. Based on this result, it is shown that the integer powers of matrices 𝑋 that satisfy the condition 𝑋^2 − 𝑉𝑛𝑋 + (−𝑞) 𝑛 𝐼 = 𝟎 for some integer 𝑛 can be found with the help of the generalized Fibonacci sequence {𝑈𝑛(𝑝, 𝑞)}. Then, some algebraic results of generalized Fibonacci and Lucas sequences are obtained using 2 × 2 dimensional special matrices. In the second section of the fourth chapter, finding the roots and Moore-Penrose inverses of matrices with the help of number sequences is studied. First, it is shown that the positive integer powers of the matrices 𝑋 satisfying the condition 𝑋^3 – 𝑉𝑛𝑋^2 + (−𝑞) 𝑛𝑋 = 𝟎 can be obtained with the help of the sequence {𝑈𝑛(𝑝, 𝑞)}. Then the relation between matrix roots and the generalized Fibonacci sequence {𝑈𝑛(𝑝, 𝑞)} is shown. The result used to find the 𝑛th root of square matrices 𝑋 that satisfy the condition 𝑋^3 – 𝑉𝑛𝑋^2 + (−𝑞)^𝑛𝑋 = 𝟎 is obtained. Then, the inverses of the matrices 𝑋 satisfying the condition 𝑋^3 – 𝑉𝑛𝑋^2 + (−𝑞)^𝑛𝑋 = 𝟎 are studied. Such matrices can be singular or nonsingular. In other words, we can not talk about the existence of inverses of all such matrices. However, we can talk about the Moore-Penrose inverses of all such matrices. First, a result is obtained to find the Moore-Penrose inverses of 𝑋, where 𝑋 is a square real matrix with 𝑋^2 − 𝑉𝑛𝑋 is symmetric and 𝑋^3 – 𝑉𝑛𝑋^2 + (−𝑞)^𝑛𝑋 = 𝟎. Then, based on this result, a way is obtained to find the Moore-Penrose inverses of non-symmetric or non-square matrices under appropriate conditions. Using the obtained results, Moore-Penrose inverses of some matrices are obtained. In the fifth chapter, some relations between the 𝑘-generalized Fibonacci sequence and matrices were obtained with a similar idea, based on the methods obtained in the previous chapters. It is known that the characteristic equation of the 𝑘-generalized Fibonacci sequence is 𝑥^𝑘 – 𝑥^𝑘−1 – 𝑥^𝑘−2−. . . −1 = 0, where k is an integer with 𝑘 ≥ 2. Based on this equation, the equation (𝑥^𝑘 – 𝑥^𝑘−1 – 𝑥^𝑘−2−. . . −1)(𝑥 − 𝑡) = 0 is obtained, where 𝑡 is a nonzero real number with 𝑡^𝑘 – 𝑡^𝑘−1 – 𝑡^𝑘−2−. . . −1 ≠ 0. The powers of the matrices 𝑋 that satisfy the resulting equation, that is, the condition 𝑋^𝑘+1 − (𝑡 + 1)𝑋^𝑘 + 𝑡𝐼 + ∑ (𝑡 − 1)𝑋^i= 𝟎, are examined. It is shown that the positive integer powers of such matrices can be obtained with the help of the 𝑘- generalized Fibonacci sequence. Then, a new sequence {𝐹(𝑛)(𝑘,𝑡)} related to the 𝑘- generalized Fibonacci sequence is defined. It is shown that the positive integer powers of matrices 𝑋 satisfying the condition 𝑋^𝑘+1 − (𝑡 + 1)𝑋^𝑘 + 𝑡𝐼 + ∑ (𝑡 − 1)𝑋^i= 𝟎 can be obtained with the help of the sequence {𝐹(𝑛)(𝑘,𝑡)}. Using the obtained results, some relations between the k -generalized Fibonacci sequence and the sequence {𝐹(𝑛)(𝑘,𝑡)} are obtained. Then, the sequence {𝐹(𝑛)(𝑘,1)} is specifically studied, and some relationships between this sequence and the 𝑘-generalized Fibonacci sequence are obtained. Additionally, using the obtained results, an explicit formula to find the 𝑛th terms of the 𝑘-generalized Fibonacci sequence is obtained, where 𝑛 is an integer with 𝑛 ≥ 𝑘. In the last chapter, the results obtained in the previous chapters are evaluated. Firstly, the results obtained throughout the study are explained. Then, some suggestions are made for future studies regarding the results obtained in the study.