Related papers: Integration Formulas Involving Fibonacci and Lucas…
We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.
The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A…
Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions;…
A convolution approach leading to an explicit computation of a value of a 4F3 function is outlined. We also investigate about the role of the dilogarithm reflection formula, leading to a remarkable consequence: in some cases, values of 4F3…
By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and…
Let $T_{k}$ be the $k^{\textrm{th}}$ Tribonacci number and $L_{n}$ be the $n^{\textrm{th}}$ Lucas number defined by their respective recurrence relation $T_{k}=T_{k-1}+T_{k-2}+T_{k-3}$ and $L_{n}=L_{n-1}+L_{n-2}$. In this study, we solve…
The following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second…
We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these…
In this article we provide with combinatorial proofs of some recent identities due to Sury and McLaughlin. We show that, the solution of a general linear recurrence with constant coefficients can be interpreted as a determinant of a matrix.…
We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.
We generalize techniques of Addison to a vastly larger context. We obtain integral representations in terms of the first periodic Bernoulli polynomial for a number of important special functions including the Lerch zeta, polylogarithm,…
We study formal power series which can be interpreted as interpolations of Fibonacci and Lucas polynomials with even (or odd) indices.
We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage…
In this paper, we study a structured family of matrices whose entries are given by products of $k$-Fibonacci and $k$-Lucas numbers. For this family, we obtain explicit and unified formulas for several classical matrix invariants, including…
The aim of this paper is to make the clarification of images faster by the formula that Franciszekn made for matrices integrations and this made Sukhvinders Algorithm complicate and slower. This paper uses the Fibonacci number to determine…
Following Bridgeman, we demonstrate several families of infinite dilogarithm identities associated with Fibonacci numbers, Lucas numbers, convergents of continued fractions of even periods, and terms arising from various recurrence…
We describe an algorithmic reduction of the search for integral points on a curve y^2 = ax^4 + bx^2 + c with nonzero ac(b^2-4ac) to solving a finite number of Thue equations. While existence of such reduction is anticipated from arguments…
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.
In \cite{Oz}, M. \"Ozdemir defined a new non-commutative number system called hybrid numbers. In this paper, we define the hybrid Fibonacci and Lucas numbers. This number system can be accepted as a generalization of the complex…