Related papers: On the Harmonic and Hyperharmonic Fibonacci Number…
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…
Fibonacci codes are self-synchronizing variable-length codes that are proven useful for their robustness and compression capability. Asymptotically, these codes provide better compression efficiency as the order of the underlying Fibonacci…
Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several…
This paper studies eight families of infinite series involving hyperbolic functions. Under some conditions, these series are linear combinations of derivatives of Eisenstein series. The paper gives a systematic method for computing the…
In this paper, we obtain some formulae for harmonic sums, alternating harmonic sums and Stirling number sums by using the method of integral representations of series. As applications of these formulae, we give explicit formula of several…
Harmonic maps are nonlinear extensions of harmonic functions. They are critical points of natural energy functionals between Riemannian manifolds. Such type of problems appear in Physics, Geometry of Finance and the study of regularity and…
This paper describes a method of calculating the transforms, currently obtained via Fourier and reverse Fourier transforms. The method allows calculating efficiently the transforms of a signal having an arbitrary dimension of the digital…
The theory of harmonic based function is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.
We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently…
We study the equation $F_n + F_m = y^p$, where $F_n$ and $F_m$ are respectively the $n$-th and $m$-th Fibonacci numbers and $p \ge 2$. We find all solutions under the assumption $n \equiv m \pmod{2}$.
Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by the recurrence $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several generalizations of this sequence and also several interesting identities. In this…
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of summands for integers in $[F_n,…
We show various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.
In this paper, we provide new applications of Fibonacci and Lucas numbers. In some circumstances, we find algebraic structures on some sets defined with these numbers, we generalize Fibonacci and Lucas numbers by using an arbitrary binary…
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…
In this work, we made a generalization that includes all bicomplex Fibonacci-like numbers such as; Fibonacci, Lucas, Pell, etc.. We named this generalization as bicomplex Horadam numbers. For bicomplex Fibonacci and Lucas numbers we gave…
The Fibonacci polynomials are defined recursively as $f_{n}(x)=xf_{n-1}(x)+f_{n-2}(x)$, where $f_0(x) = 0$ and $f_1(x)= 1$. We generalize these polynomials to an arbitrary number of variables with the $r$-Fibonacci polynomial. We extend…
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.…
In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations…
Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of…