• Kyungpook Mathematical Journal
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• v.57 no.1
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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.

• Honam Mathematical Journal
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• v.40 no.1
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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.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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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.

• Honam Mathematical Journal
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• v.44 no.4
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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.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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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.

• Honam Mathematical Journal
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• v.40 no.1
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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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• v.24 no.4
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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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• v.23 no.3
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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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• v.22 no.3
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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.

• School Mathematics
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• v.12 no.4
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• pp.619-638
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• 2010

In this paper, we listed and reviewed some properties on polygonal numbers, pyramidal numbers and Pascal's triangle, and Fibonacci sequence. We discussed that the properties of gnomonic numbers, polygonal numbers and pyramidal numbers are explained integratively by introducing the generalized k-dimensional pyramidal numbers. And we also discussed that the properties of those numbers and relationships among generalized k-dimensional pyramidal numbers, Pascal's triangle and Fibonacci sequence are suitable for teaching and learning of mathematical reasoning and connections.