Kunle Adegoke
By applying the classic telescoping summation formula and its variants to identities involving inverse hyperbolic tangent functions having inverse powers of the golden ratio as arguments and employing subtle properties of the Fibonacci and Lucas numbers, we derive interesting general infinite product identities involving these numbers.
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Fibonacci or Lucas numbers that are products of two Lucas numbers or two Fibonacci numbers
本文研究 Fibonacci 数与 Lucas 数互为两项乘积的丢番图方程，利用 Dujella–Pethö 引理和 Matveev 定理求出全部解及公共项。
We derive generalizations of a couple of inverse tangent summation identities involving Fibonacci and Lucas numbers. As byproducts we establish many new inverse tangent identities involving the Fibonacci and Lucas numbers.
In this study, we define a new type of Fibonacci and Lucas num- bers which are called bicomplex Fibonacci and bicomplex Lucas numbers. We obtain the well-known properties e.g. Docagnes, Cassini, Catalan for these new types. We also give the identities of negabicomplex Fibonacci and negabi- complex Lucas numbers, Binet formulas and relations of them.
We derive closed form expressions for finite and infinite Fibonacci-Lucas sums having products of Fibonacci or Lucas numbers in the denominator of the summand. Our results generalize and extend those obtained by pioneer Brother Alfred Brousseau and later researchers.
By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio, $α=(1+\sqrt 5)/2$ and its inverse, $β=-1/α=(1-\sqrt 5)/2$, a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, we follow the reverse course: we derive numerous Fibonacci and Lucas identities by making use of the well-known expressions for the powers of $α$ and $β$ in terms of Fibonacci and Lucas numbers.
In this paper, we define the incomplete h(x)-Fibonacci and h(x)-Lucas polynomials, we study recurrence relations and some properties of these polynomials
We define the convolved h(x)-Fibonacci polynomials as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the h(x)-Fibonacci and h(x)-Lucas polynomials. Moreover we obtain the convolved h(x)-Fibonacci polynomials form a family of Hessenberg matrices.
In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas numbers and generalized order-k Fibonacci numbers. In addition, we obtain Binet formula and combinatorial representation for generalized order-k Lucas numbers by using properties of generalized Fibonacci numbers.
A second order polynomial sequence is of Fibonacci type (Lucas type) if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers. In this paper we generalize identities from Fibonacci numbers and Lucas numbers to Fibonacci type and Lucas type polynomials. A Fibonacci type polynomial is equivalent to a Lucas type polynomial if they both satisfy the same recurrence relations. Most of the identities provide relationships between two equivalent polynomials. In particular, each type of identities in this paper relate the following polynomial sequences: Fibona…
Powers of Fibonacci polynomials are expressed as single sums, improving on a double sum recently seen in the literature.
At this paper, we derive some relationships between permanents of one type of lower-Hessenberg matrix and the Fibonacci and Lucas numbers and their sums.
Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular cases of the more general formulas derived here.
In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations.
In this paper, firstly, we define the Qq-generating matrix for bi-periodic Fibonacci polynomial. And we give nth power, determinant and some properties of the bi-periodic Fibonacci polynomial by considering this matrix representation. Also, we introduce the Hadamard products for bi-periodic Fibonacci Qnq generating matrix and bi-periodic Lucas Qnl generating matrix of which entries is bi-periodic Fibonacci and Lucas numbers. Then, we investigate some properties of these products.
In this paper, we introduce the bi-periodic Lucas matrix sequence and present some fundamental properties of this generalized matrix sequence. Moreover, we investigate the important relationships between the bi-periodic Fibonacci and Lucas matrix sequences. We express that some behaviours of bi-periodic Lucas numbers also can be obtained by considering properties of this new matrix sequence. Finally, we say that the matrix sequences as Lucas, $k$-Lucas and Pell-Lucas are special cases of this generalized matrix sequence.
We show that the product of two consecutive Fibonacci (respectively Lucas) numbers is divisible by the sum of their indices if this sum is a prime number different from 5 and in the form (4r+1)(respectively (4r+3)).
We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of the form \mbox{$\sum_{k = 1}^n {F_{2rk}^3 }$} and \mbox{$\sum_{k = 1}^n {L_{2rk}^3 }$}, where $r$~and~$n$ are integers.
A supercongruence fantasy on Fibonacci and Lucas (after Guillera)
本文在吉拉尔的基础上，将拉马努金型超同余推广到更高级层次，使斐波那契、卢卡斯等序列自然出现。
Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ the $n$-th Lucas number. We completely solve the family of cubic Thue equations $${(X-F_nY)(X-L_nY)X-Y^3=\pm1}$$ and show that there are no non-trivial solutions for $n\neq 1,3$.
We obtain closed-form expressions for all sums of the form \mbox{$\sum_{k = 1}^n {F_{mk}{}^4 }$} and \mbox{$\sum_{k = 1}^n {L_{mk}{}^4 }$} and their alternating versions, where $F_i$ and $L_i$ denote Fibonacci and Lucas numbers respectively. Our results complement those of Melham who studied the alternating sums.
A second order polynomial sequence is of Fibonacci-type $\mathcal{F}_{n}$ (Lucas-type $\mathcal{L}_{n}$) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Under certain conditions these polynomials are irreducible if and only if $n$ is a prime number. For example, the Fibonacci polynomials, Pell polynomials, Fermat polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials are irreducible when $n$ is a prime number; and Chebyshev polynomials (second kind), Morgan-Voyce polynomials (Fibonacci type), and Vieta polynomials are reducible when…
Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of $ a $ and $ b $, we mean the both concatenations $\overline{ab}$ and $\overline{ba}$ together, where $ a $ and $ b $ are any two non negative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations $ F_n=10^d F_m +L_k $ and $ F_n=10^d L_m+F_k $ in non-negative integers $ (n,m,k) ,$ where $ d $ d…
Inverses of Fibonacci and Lucas Numbers via Rational Indices
研究通过分母函数求有理指数的斐波那契数和卢卡斯数的逆，推导\(F_X\)的一般显式公式，确定其与卢卡斯数及斐波那契序列的关系，计算相关数的单位，证明乘法逆元存在，揭示序列新结构性质。
The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.
In this study, we define the dual Fibonacci quaternion and the dual Lucas quternion. We derive the relations between the dual Fibonacci and the dual Lucas quaternion which connected the Fibonacci and the Lucas numbers. Furthermore, we give the Binet and Cassini formulas for these quaternions.
A common generalization of Dickson polynomials, Fibonacci polynomials, and Lucas polynomials and applications
本文定义了满足线性递推关系的更一般的多变量多项式族，给出了显式公式和行列式表达式，并将其应用于2阶递推多项式，揭示了Fibonacci、Lucas和Dickson多项式等的关系与恒等式，是对多项研究的统一推广。
On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers
本文研究了Fibonacci和Lucas数的混合b-拼接形式，通过求解两个丢番图方程确定哪些Lucas数可表示为Fibonacci和Lucas数的拼接，证明仅存在有限多个这样的Lucas数。
By means of two complex-valued functions (depending on an integer parameter P>=1) we construct helices of integer ratio R>=1 related to the so-called Binet formulae for P-Lucas and P-Fibonacci sequences. Based on these functions a new map is defined and we show that its three-dimensional representation is also a helix. After proving that the lattice points of these later helix satisfy certain diophantine Pell's equations we call it a Pell's helix. We prove that for P-Fibonacci and Pell's helices the respective ratio is an invariant, contrasting to the P-Lucas helices whose ratio depends on P. …
A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials. The \emph{resultant} of two polynomials is the determinant of the Sylvester matrix and the \emph{discriminant} of a polynomial $p$ is the resultant of $p$ and its derivative. We study …
Two Primes and Five Mixed Fibonacci-Lucas Summands
该研究用圆法证明，对u=1至4，每个足够大的正整数可表为两素数加u个斐波那契数与5−u个卢卡斯数之和，项可重复且递推指标为正。
Let $(U_{n}(P,Q) $ and $(V_{n}(P,Q) $ denote the generalized Fibonacci and Lucas sequence, respectively. In this study, we assume that $Q=1.$ We determine all indices $n$ such that $U_{n}=5\square $ and $U_{n}=5U_{m}\square $ under some assumptions on $P.$ We show that the equation $V_{n}=5\square $ has the solution only if $n=1$ for the case when $% P$ is odd. Moreover, we show that the equation $V_{n}=5V_{m}\square $ has no solutions.
We construct the sequences of Fibonacci and Lucas at any quadratic field $\mathbb{Q}(\sqrt{d}\ )$ with $d>0$ square free, noting in general that the properties remain valid as those given by the classical sequences of Fibonacci and Lucas for the case $d = 5$, under the respective variants. For this construction, we use the fundamental unit of $\mathbb{Q}(\sqrt{d}\ )$ and then we observe the generalizations for any unit of $\mathbb{Q}(\sqrt{d}\ )$ where, under certain conditions, some of this constructions correspond to $k$-Fibonacci sequence for some $k\in \mathbb{N}$. Of course, for both sequ…
Let $α=(1+\sqrt 5)/2$, the golden ratio, and $β=-1/α=(1 - \sqrt 5)/2$. Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, defined by $F_n=(α^n -β^n)/\sqrt 5$ and $L_n=α^n + β^n$, for all non-negative integers. We derive base~$α$ expansions of $\log F_n$, $\log L_n$, $\arctan\dfrac1{F_n}$ and $\arctan\dfrac1{L_n}$ for all positive integers $n$.
Recall that repdigit in base $g$ is a positive integer that has only one digit in its base $g$ expansion, i.e. a number of the form $a(g^m-1)/(g-1)$, for some positive integers $m\geq 1$, $g\geq 2$ and $1\leq a\leq g-1$. In the present study we investigate all Fibonacci or Lucas numbers which are expressed as products of three repdigits in base $g$. As illustration, we consider the case $g=10$ where we show that the numbers 144 and 18 are the largest Fibonacci and Lucas numbers which can be expressible as products of three repdigits respectively. All this can be done using linear forms in loga…
In this note, we obtain some identities for the generalized Fibonacci polynomial by using the Q(x) matrix. These identities including the Cassini identity and Honsberger formula can be applied to some polynomial sequences, such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials, Fermat polynomials, Fermat-Lucas polynomials, and so on.
Monogenity of the irreducible factors of $(k,t)$-Fibonacci and $(k,t)$-Lucas polynomials
本文利用 Dickson 多项式与分圆域的联系，证明了广义 $(k,t)$-Fibonacci 和 $(k,t)$-Lucas 多项式不可约因子生成数域的单项性，并推广至经典 Fibonacci、Lucas 及 Chebyshev 多项式。
In this study, we find continued fraction expansion of sqrt(d) when d=a^2+2a where a is positive integer. We consider the integer solutions of the Pell equation x^2-(a^2+2a)y^2=N when N={-1,+1,-4,+4}. We formulate the n-th solution (x_{n},y_{n}) by using the continued fraction expansion. We also formulate the n-th solution (x_{n},y_{n}) via the generalized Fibonacci and Lucas sequences.
In 2007, Tachiya gave necessary and sufficient conditions for the transcendence of certain infinite products involving Fibonacci numbers $F_k$ and Lucas numbers $L_k$. In the present note, we explicitly evaluate two classes of his algebraic examples. Special cases are$$\prod_{n=1}^{\infty}(1+\frac{1}{F_{2^n+1}})=\frac{3}ϕ, \qquad \prod_{n=1}^{\infty}(1+\frac{1}{L_{2^n+1}})=3-ϕ\, ,$$where $ϕ=(1+\sqrt{5})/2$ is the golden ratio.
Monogenity of Fibonacci polynomials and Lucas polynomials
研究Fibonacci多项式F_n(x)和Lucas多项式L_n(x)的不可约因子的单生成性，证明对所有奇数n，F_n(x)的不可约因子是单生成的；对所有偶数n，L_n(x)的不可约因子是单生成的。
On Diophantine equations with Fibonacci and Lucas sequences associated to Hardy-Ramanujan problem
该研究围绕哈代-拉马努金问题，探究涉及斐波那契序列和卢卡斯序列的两类三次丢番图方程，找出了兼具哈代-拉马努金数属性的斐波那契数与卢卡斯数。
This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.
We explicitly solve the diophantine equations of the form $$ A_{n_1}A_{n_2}\cdots A_{n_k}\pm 1 = B_m^2 $$ where $(A_n)_{n\geq 0}$ and $(B_m)_{m\geq 0}$ are either the Fibonacci sequence or Lucas sequence. This extends the result of D. Marques (2011) and L. Szalay (2012) concerning a variant of Brocard-Ramanujan equation. This is a manuscript of the article published in Communications of the Korean Mathematical Society, 32(3) (2017), pp. 511-522
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.
Based on well-known properties of Fibonacci and Lucas numbers and polynomials we give a self-contained approach to some bivariate analogs.
In this study, we find continued fraction expansion of sqrt(d) when d=a^2b^2-b and d=a^2b^2-2b where a and b are positive integers. We consider the integer solutions of the Pell equations x^2-(a^2b^2-b)y^2=N and x^2-(a^2b^2-2b)y^2=N when N is {+-1,+-4}. We formulate the n-th solution (x_{n},y_{n}) by using the continued fraction expansion. We also formulate the n-th solution (x_{n},y_{n}) in terms of generalized Fibonacci and Lucas sequences.
Fibonacci Numbers and Their Lucas Coefficients
本文建立 Fibonacci 数与 Lucas 数之间的卷积恒等式，将 Fn 表示为 Lucas–Fibonacci 乘积和除以 n-1，并通过归纳法与数值示例加以验证。
In 1989, Ming Luo \cite{L2} showed that the Fibonacci number $U_n$ is Triangular if and only if $n=\pm1,2,4,8,10$. For this, he established a Jacobi Symbol Criterion. Moreover, he observed that this problem is equivalent to finding all integer points on two elliptic curves. In this paper, we prove a Jacobi Symbol Criterion for more general families of binary recurrences. In addition, applying the criterion and elementary methods, we determine all integer points on the elliptic curves $y^2=5x^2(x+3)^2+4(-1)^n$.
Some power series extensions of Fibonacci and Lucas polynomials
本研究聚焦形式幂级数领域，探讨可作为偶/奇下标斐波那契、卢卡斯多项式插值的形式幂级数，拓展了相关多项式的幂级数研究方向。
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
The local Dold congruence for Fibonacci and Lucas sequences
本文刻画 Fibonacci 与 Lucas 序列中满足 Dold 同余或可实现性的素数，计算修复乘子并给出四类素数集的密度，分别为 (3/4,1/3,1,1/3) 与 (1/2,1/2,1,1)。
In this paper we present two families of Fibonacci-Lucas identities, with the Sury's identity being the best known representative of one of the families. While these results can be proved by means of the basic identity relating Fibonacci and Lucas sequences we also provide a bijective proof. Both families are then treated by generating functions.
The Discriminant of the Characteristic Polynomial of the $k$th Fibonacci sequence is not a member of the $k$th Lucas sequence
本文研究了$k$次Lucas序列的数论性质，证明了$k$次Fibonacci序列特征多项式的判别式不属于$k$次Lucas序列。
On balancing and Lucas-balancing numbers expressible as product of two $k$-Fibonacci numbers
本文研究数论中可表示为两个k-广义斐波那契数乘积的平衡数与Lucas-平衡数，完整刻画了这类数的全部情形。
In this paper, we establish several formulae for sums and alternating sums of products of generalized Fibonacci and Lucas numbers. In particular, we recover and extend all results of Z. Cerin and Z. Cerin & G. M. Gianella, more easily.
In this paper, we give two new coding algorithms by means of right circulant matrices with elements generalized Fibonacci and Lucas polynomials. For this purpose, we study basic properties of right circulant matrices using generalized Fibonacci polynomials $F_{p,q,n}\left( x\right) $, generalized Lucas polynomials $L_{p,q,n}\left( x\right) $ and geometric sequences.
The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the new sequence S_{k,n} with initial conditions S_{k,0} = 2b and S_{k,1} = bk + a, which is generated by the recurrence relation S_{k,n} = kS_{k,n-1} +S{k,n-2} for n >= 2, where a, b, k are real numbers. Using the sequence S_{k,n}, we introduce and prove some special identities. Also, we deal with the circulant and skew circulant matrices for the sequence S_{k,n}.
The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers and genera.
In this paper, we consider a generalization of Horadam sequence fwng which is defined by the recurrence relation wn = x(n)wn-1+ cwn-2; where x(n) = a if n is even, x(n) = b if n is odd with arbitrary initial conditions w0;w1 and nonzero real numbers a, b and c. As a special case, by taking initial conditions 0, 1 and 2, b we define the sequences {un} and {vn}, respectively. The main purpose of this study is to derive some basic properties of the sequences {un}, {vn} and {wn} by using matrix approach.
We introduce the notion of Fibonacci and Lucas derivations of the polynomial algebras and prove that any element of kernel of the derivations defines a polynomial identity for the Fibonacci and Lucas polynomials. Also, we prove that any polynomial identity for Appel polynomial yields a polynomial identity for the Fibonacci and Lucas polynomials and describe the corresponding intertwining maps.
We continue our study on relationships between Bernoulli polynomials and balancing (Lucas-balancing) polynomials. From these polynomial relations, we deduce new combinatorial identities with Fibonacci (Lucas) and Bernoulli numbers. Moreover, we prove a special identity involving Bernoulli polynomials and Fibonacci numbers in arithmetic progression. Special cases and some corollaries will highlight interesting aspects of our findings. Our results complement and generalize these of Frontczak (2019).
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