• Kyungpook Mathematical Journal
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• 제57권1호
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• pp.1-84
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• 2017

In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for $F_p$, where p is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, m and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.

• 호남수학학술지
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• 제40권1호
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• pp.199-210
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• 2018

In this paper we obtain some formulae for several sums of generalized Fibonacci numbers $U_n$ and generalized Lucas numbers $V_n$ and their dual forms $G_n$ and $H_n$ by using extensions of an interesting identity by A. R. Amini for Fibonacci numbers to these four kinds of generalizations and their first and second derivatives.

• 대한수학회지
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• 제48권1호
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• pp.33-48
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• 2011

The notion of the generalized Fibonacci matrix $\mathcal{F}_n^{(a,b,s)}$ of type s, whose nonzero elements are generalized Fibonacci numbers, is introduced in the paper [23]. Regular case s = 0 is investigated in [23]. In the present article we consider singular case s = -1. Pseudoinverse of the generalized Fibonacci matrix $\mathcal{F}_n^{(a,b,-1)}$ is derived. Correlations between the matrix $\mathcal{F}_n^{(a,b,-1)}$ and the Pascal matrices are considered. Some combinatorial identities involving generalized Fibonacci numbers are derived. A class of test matrices for computing the Moore-Penrose inverse is presented in the last section.

• 호남수학학술지
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• 제44권4호
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• pp.473-484
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• 2022

In this study, some new relations between generalized Fibonacci numbers and matrices are given. The work is designed in three stages: Firstly, it is obtained a relation between generalized Fibonacci numbers and integer powers of the matrices X satisfying the relation X² = pX +qI, and also, many results are derived from obtained relation. Then, it is established more general relation between generalized Fibonacci numbers and the square matrices X satisfying the condition X² = V_nX - (-q)ⁿI. Finally, some applications and numerical examples related to the obtained results are given.

• 대한수학회지
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• 제45권2호
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• pp.479-491
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• 2008

In [4], the authors studied the Pascal matrix and the Stirling matrices of the first kind and the second kind via the Fibonacci matrix. In this paper, we consider generalizations of Pascal matrix, Fibonacci matrix and Pell matrix. And, by using Riordan method, we have factorizations of them. We, also, consider some combinatorial identities.

• 호남수학학술지
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• 제40권1호
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• pp.139-153
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• 2018

Let $P{\geq}3$ be an integer and let ($U_n$) denote generalized Fibonacci sequence defined by $U_0=0$, $U_1=1$ and $U_{n+1}=PU_n-U_{n-1}$ for $n{\geq}1$. In this study, when P is odd, we solve the equation $U_n=11x^2+1$. We show that only $U_1$ and $U_2$ may be of the form $11x^2+1$.

• Korean Journal of Mathematics
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• 제24권4호
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• pp.681-691
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• 2016

In this note, we consider a generalized Fibonacci sequence {$q_n$}. We give a generating matrix for {$q_n$}. With the aid of this matrix, we derive and re-prove some properties involving terms of this sequence.

• Korean Journal of Mathematics
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• 제23권3호
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• pp.371-377
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• 2015

In this note, we consider a generalized Fibonacci sequence {$q_n$}. Then give a connection between the sequence {$q_n$} and the Chebyshev polynomials of the second kind $U_n(x)$. With the aid of factorization of Chebyshev polynomials of the second kind $U_n(x)$, we derive the complex factorizations of the sequence {$q_n$}.

• Korean Journal of Mathematics
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• 제22권3호
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• pp.443-454
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• 2014

In this paper, we give linearization of generalized Fi-bonacci sequences {$g_n$} and {$q_n$}, respectively, defined by Eq.(5) and Eq.(6) below and use this result to give the matrix form of the nth power of a companion matrix of {$g_n$} and {$q_n$}, respectively. Then we re-prove the Cassini's identity for {$g_n$} and {$q_n$}, respectively.

• 대한수학교육학회지:학교수학
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• 제12권4호
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• pp.619-638
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• 2010

본 고에서는 평면이나 공간에서 정의된 도형수가 일반적으로 유한 차원에서 일반화될 때 저차원의 도형수인 그노몬수, 다각수 그리고 다각뿔수의 성질을 통합적으로 설명할 수 있음을 논하고, 도형수와 파스칼 삼각형, 피보나치 수열의 성질과 그들 사이의 관계를 알아봄으로써 이들에 대한 성질 탐구가 수학적 추론과 연결성을 지도하기 적합한 소재가 될 수 있음을 논한다.