Bi-Periodic Fibonacci Matrix Sequences

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Thesis Type: Postgraduate

Institution Of The Thesis: Yıldız Technical University, Graduate School of Natural and Applied Sciences, MATEMATİK ANABİLİM DALI, Turkey

Approval Date: 2022

Thesis Language: Turkish

Student: ELİF TÜYSÜZ

Supervisor: Gürsel Yeşilot

Open Archive Collection: AVESIS Open Access Collection

Abstract:

This thesis consists of 5 chapters. The first chapter, some basic information about Fibonacci numbers is given. Leonardo Fibonacci and the rabbit problem are mentioned. Afterwards, a literature review is given. Finally, the objective of the thesis and the hypothesis are given. In the second chapter, the definitions of the Binet formula, Catalan identity and Cassini identity, which will be used in the thesis, are given. Also, the definition of Q-matrix is given. In the third chapter, the definition of bi-periodic Fibonacci sequences is given. The generating function of bi-periodic Fibonacci sequences is expressed and the finding of this function is proved. Binet formula for bi-periodic Fibonacci sequences is given and its proof is done. The Catalan identity is given and its proof is made with the Binet formula. Cassini identity is given. The d'Ocagne identity is given and its proof is made using the Binet formula. Finally, the sums involving the binomial coefficients are given, and the proof of these sums is made using the Binet formula. In the fourth chapter, bi-periodic Fibonacci matrix sequence is defined. The general term n. of this sequence is given and its discovery is proved by induction. The general term is given with a determinant and its proof is done using induction. According to these two theorems, a result has been given and its proof has been made. The generator function of the bi-periodic Fibonacci matrix sequence is given and its proof is made. Binet's formula is given for the bi-periodic Fibonacci matrix sequence. Finally, the general sum of bi-periodic Fibonacci matrix sequences are given. While calculating this general sum, Binet's formula was used. In the last chapter of thesis, conclusion and suggestions are included.