The paper presents, among others, the golden number $\varphi$ as the limit of the quotient of neighboring terms of the Fibonacci and Fibonacci type sequence by means of a fixed point of a mapping of a certain interval with the help of Edelstein’s theorem. To demonstrate the equality  , where $f_n$ is $n$-th Fibonacci number also the formula from Corollary has been applied. It was obtained using some relationships between Fibonacci and Lucas numbers, which were previously justified.

Integer sequences of the form $\lfloor n^c\rfloor$, where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα+β⌋ in a very good way. Following this approach, we are led to an estimate of the difference ${\sum }_{n\le x}\phi \left(\lfloor n^c\rfloor \right)-1/c{\sum }_{n\le x^c}\phi \left(n\right)n^{1/c-1}$, which measures the deviation of the mean value of φ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by $\lfloor n^c\rfloor$ attains both of its values with...

We say a sequence $={\left(sₙ\right)}_{n\ge 0}$ is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, ${}_a={\left(uₙ\right)}_{n\ge 0}$, defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences ${}_a\pm k$ are simultaneously primefree. This...

We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0,∞)² → (0,∞) be a continuous function such that (a) 0 < f(x,y) < px + (1-p)y for some p ∈ (0,1) and for all x,y ∈ (0,α), where α > 0; (b) $f(x,y)=px+(1-p)y-{\sum }_{s=m}^{\infty }s(x,y)$ uniformly in a neighborhood of the origin, where m > 1, ${}_s(x,y)={\sum }_{i=0}^s{a}_{i,s}{x}^{s-i}{y}^i$; (c) $ₘ(1,1)={\sum }_{i=0}^m{a}_{i,m}>0$. Let x₀,x₁ ∈ (0,α) and $x_{n+1}=f(xₙ,x_{n-1})$, n ∈ ℕ. Then the sequence (xₙ) satisfies the following asymptotic formula: $xₙ\sim {((2-p)/\left((m-1){\sum }_{i=0}^m{a}_{i,m}\right))}^{1/(m-1)}1/\sqrt[m-1]{n}$.

A generalized form of the usual Lognormal distribution, denoted with ${}_{\gamma }$, is introduced through the γ-order Normal distribution ${}_{\gamma }$, with its p.d.f. defined into (0,+∞). The study of the c.d.f. of ${}_{\gamma }$ is focused on a heuristic method that provides global approximations with two anchor points, at zero and at infinity. Also evaluations are provided while certain bounds are obtained.

Melham discovered the Fibonacci identity $F_{n+1}{F}_{n+2}{F}_{n+6}-F{³}_{n+3}=(-1)ⁿFₙ$. He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and $Wₙ=p{W}_{n-1}+q{W}_{n-2}$ and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: $W_{n+1}{W}_{n+2}{W}_{n+6}-W{³}_{n+3}=e{q}^{n+1}(p³W_{n+2}-q²W_{n+1})$. There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $FₙF_{n+4}{F}_{n+5}-F{³}_{n+3}={(-1)}^{n+1}{F}_{n+6}$. We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is $WₙW_{n+4}{W}_{n+5}-W{³}_{n+3}=eqⁿ(p³W_{n+4}-q{W}_{n+5})$.

Characterizations of equicontinuity and convergent sequences are given for the space ${}_C^{\text{'}}\left(ℝⁿ\right)$ of rapidly decreasing distributions and the space ${}_M\left(ℝⁿ\right)$ of slowly increasing infinitely differentiable functions.

We prove the existence of a limit distribution of the normalized well-distribution measure $W\left(E_N\right)/\sqrt{N}$ (as $N\to \infty$) for random binary sequences $E_N$, by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.

Let $k\ge 2$ and define $F^{\left(k\right)}:={\left(F_n^{\left(k\right)}\right)}_{n\ge 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{\left(k\right)}=F_{n-1}^{\left(k\right)}+F_{n-2}^{\left(k\right)}+\cdots +F_{n-k}^{\left(k\right)}$, with initial conditions $0,0,\cdots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{\left(k\right)}=1$. The sequences $F:=F^{\left(2\right)}$ and $T:=F^{\left(3\right)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^{\left(k\right)}=F_m^{\left(\ell \right)}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{\left(k\right)}\cap F^{\left(m\right)}$ which gives...

It is proved that the solution to the initial value problem ${\partial }_{t}u=\partial {²}_{x}u+u²$, u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^s$ in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.

We derive two identities for multiple basic hyper-geometric series associated with the unitary $U(n+1)$ group. In order to get the two identities, we first present two known $q$-exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two $U(n+1)$ $q$-Chu-Vandermonde summations established by Milne, we arrive at our results. Using the identities obtained in this paper, we give two interesting identities involving binomial...

In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p-$Laplacian, where $1<p<2$. We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t\to \infty$.

Our main purpose is to establish the existence of a positive solution of the system ⎧$-{∆}_{p\left(x\right)}u=F(x,u,v)$, x ∈ Ω, ⎨$-{∆}_{q\left(x\right)}v=H(x,u,v)$, x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded domain with C² boundary, $F(x,u,v)={\lambda }^{p\left(x\right)}[g\left(x\right)a\left(u\right)+f\left(v\right)]$, $H(x,u,v)={\lambda }^{q(x})[g\left(x\right)b\left(v\right)+h\left(u\right)]$, λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $-{∆}_{p\left(x\right)}u=-{div\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

CONTENTS1. Introduction.............................................................................................................................................52. Assumptions and notations................................................................................................................63. An important asymptotic distribution..................................................................................................84. The asymptotic distribution of $K{*}_{\mu ,\nu }$ under the...

A generalization of the well-known Fibonacci sequence ${Fₙ}_{n\ge 0}$ given by F₀ = 0, F₁ = 1 and $F_{n+2}=F_{n+1}+Fₙ$ for all n ≥ 0 is the k-generalized Fibonacci sequence ${F{ₙ}^{\left(k\right)}}_{n\ge -(k-2)}$ whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula $Fₙ²+F{²}_{n+1}²=F_{2n+1}$ holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This...