• Kyungpook Mathematical Journal
• /
• v.57 no.1
• /
• pp.1-84
• /
• 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
• /
• v.40 no.1
• /
• pp.199-210
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.2
• /
• pp.535-547
• /
• 2019

In this paper, by using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and $Peth{\ddot{o}}$, we find all generalized Fibonacci numbers which are Pell numbers. This paper continues a previous work that searched for Pell numbers in the Fibonacci sequence.

• Journal of Applied and Pure Mathematics
• /
• v.6 no.1_2
• /
• pp.47-54
• /
• 2024

In this paper, we investigate the distribution of the zeros of the Bernoulli-Fibonacci polynomials by using computer.

• Honam Mathematical Journal
• /
• v.44 no.4
• /
• pp.473-484
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.29 no.3
• /
• pp.387-399
• /
• 2014

The ratios of any two Fibonacci numbers are expressed by means of semisimple continued fraction.

• Honam Mathematical Journal
• /
• v.42 no.1
• /
• pp.49-62
• /
• 2020

In this paper, we consider interesting binomial and alternating binomial triple sums evaluated in multiplication forms in terms of the Fibonacci and Lucas numbers.

• Communications of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.89-96
• /
• 2023

Edgar obtained an identity between Fibonacci and Lucas numbers which generalizes previous identities of Benjamin-Quinn and Marques. Recently, Dafnis provided an identity similar to Edgar's. In the present article we give some generalizations of Edgar's and Dafnis's identities.

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

• Kyungpook Mathematical Journal
• /
• v.61 no.3
• /
• pp.473-486
• /
• 2021

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper.