Triangular numbers and generalized fibonacci polynomial

A new efficient hierarchical multi-secret sharing scheme based on linear homogeneous recurrence relations

In the following chapter we address the techniques for the resolution of some celebrated recurrence relations. We will discuss in detail the linear recurrences with constant coefficients. Our emphasis goes to the application of the theory: the proofs, though elementary, are relegated to the end of the chapter. We proceed step by step in showing first how to solve just homogeneous recurrences, then how to find a particular solution in some special cases and only finally how to obtain all the solutions to the original problem. We also consider linear recurrences with variable coefficients and the divide and conquer recurrences: here we focus on the order of magnitude of the solutions, a fact which has an impact in the analysis of algorithms. There are about 40 examples and 50 classified problems.

This paper concerns new numerical solutions for certain coupled system of fractional differential equations through the employment of the so-called generalized Fibonacci polynomials. These polynomials include two parameters and they generalize some important well-known polynomials such as Fibonacci, Pell, Fermat, second kind Chebyshev, and second kind Dickson polynomials. The proposed numerical algorithm is essentially built on applying the spectral tau method together with utilizing a Fejer quadrature formula. For the implementation of our algorithm, we introduce a new operational matrix of fractional-order differentiation of generalized Fibonacci polynomials. A careful investigation of convergence and error analysis of the proposed generalized Fibonacci expansion is performed. The robustness of the proposed algorithm is tested through presenting some numerical experiments.

In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under the convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first and second kind. Hence, we call them Stirling operators. In order to construct matrix representations of the roots of GFPs, we use Stirling operators of the first kind. We give explicit examples to show how Stirling operators of the second kind appear in low degree cases for the WIP-roots. As a consequence of the matrix construction we have matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure.

CHAPTER 4 - Recurrence Relations and Algebraic Equations

Generalized Fibonacci Polynomials and the Functional Iteration of Rational Functions of Degree One

Background The study of bi-univalent functions has gained considerable attention due to its relevance in geometric function theory, particularly in determining bounds for the initial coefficients of analytic functions. Previous research has introduced several subclasses of bi-univalent functions using tools such as convolution, subordination, differential subordination, and various families of special polynomials. Methods Motivated by these developments, this work constructs new subclasses of bi-univalent functions by employing generalized Fibonacci polynomials. The analytic properties of these polynomials are used to define the classes and derive the corresponding functional relations needed for coefficient estimation. Results For the newly introduced subclasses, bounds for the initial coefficients are established. Furthermore, Fekete–Szegö inequalities are obtained for each subclass, extending previously known results and demonstrating the influence of the generalized Fibonacci structure on coefficient behavior. Conclusions The results contribute to the ongoing development of bi-univalent function theory by providing broader subclasses and sharper estimates. The use of generalized Fibonacci polynomials offers a flexible analytical framework that can be further applied to related classes in future studies.

In this research article, we build and implement an efficient spectral algorithm for handling linear/nonlinear mixed Volterra-Fredholm integro-differential equations. First, we expand the exact solution as a truncated series of the generalized Fibonacci polynomials, and then we discretize the equation via Simpson's quadrature formula. Finally, we collocate the resulted residual at the roots of the shifted first-kind Chebyshev polynomials. Also, the rate of convergence is studied and the truncation error estimate is reported. Some numerical examples are exhibited to prove the applicability and accuracy of the algorithm.

Recently the solution of multidimensional, linear, and homogeneous recurrence relations, or partial difference equations (PDE), was obtained via a multidimensional extension of the combinatorics function technique, developed by Antippa and Phares. Combinatorics functions of the first and second kind are representations of ’’restricted’’ paths connecting two points in an n-dimensional space. These functions are shown to give the solution of the most general linear and inhomogeneous PDE. The consistency of the PDE with the initial value conditions is also discussed. Applications of the method are given elsewhere.

Abstract. Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a − p(a)Dm−2(a)/Dm−1(a)}, where Dm(a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each Dm(a) is also representable as a Toeplitz determinant. Except possibly for the locus of points equidistant to two distinct roots, given any input a, the Basic Sequence converges to a root of p. The roots of p partition the Euclidean plane into Voronoi regions. Under some regularity assumption (e.g. simplicity of the roots), for almost all inputs within the Voronoi polygon of a root, the corresponding Basic Sequence converges to that root. The discovery of the Basic Sequence, its error estimates, and several of its properties are consequences of our previous analysis of a fundamental family of iteration functions {Bm(z)}, called the Basic Family. Given any fixed m ≥ 2 and an appropriate input a0, the fixed-point iterates ak+1 = Bm(ak) converge to a simple root having order m. The Basic Sequence corresponds to the pointwise evaluation of the Basic Family. We present several fractal images that confirm the theoretical convergence results: as m increases, the basins of attractions to the roots, as computed with respect to the iteration function Bm(z), rapidly converge to the Voronoi regions. Thus the regions with chaotic behavior rapidly shrink to the boundaries of the Voronoi regions. The interplay between the Basic Sequence and the Basic Family result in new algorithms for approximation of polynomial roots. Also as a byproduct of the analysis we give determinantal representation of solution of homogeneous linear recurrence relations, easily computable bounds on the modulus of terms of such recurrence relations, and fast computation of determinant of certain Toeplitz matrices. Finally, we prove a relationship between the Basic Sequence and the iterates of Bernoulli Method for approximation of extreme roots.

The aim of this paper is to investigate the behavior of quotients of sequences w n defined by the linear homogeneous recurrence relation $$w_{n + 2} = pw_{n + 1} - qw_n$$ (1) with arbitrary initial values w 0 = a, w 1 = b belonging to a normed field K, and p, q ∈ K. We shall write w n (a,b; p,q) when it swill be necessary to specify the initial data and parameters. In [3], Horadam has studied the properties of these sequences. If w n ≠ 0, ∀n ∈ N, then the sequence of quotien1ts $$Q_n (a,b;p,q) = \frac{{w_{n + 1} }} {{w_n }}$$ (2.1) is defined in K. If the field K is normed and α, β are the roots of the polynomial x 2 – px + q, with ∥ α ∥ > ∥ β ∥ then the sequences of Q n ’s, where it is defined, converges, and $$ \mathop{{\lim }}\limits_{{n \to \infty }} {Q_n} = \alpha. $$ (3)

In this paper, we investigate the generalized Guglielmo sequences and we deal with, in detail, four special cases, namely, triangular, triangular-Lucas, oblong and pentagonal sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

We consider the problem of computing a linear recurrence relation (or equivalently a Linear Feedback Shift Register) of minimum order for a finite sequence over a field, with the additional requirement that not only the highest but also the lowest coefficient of the recurrence is non-zero. Such a recurrence relation can then be used to generate the sequence in both directions (increasing or decreasing order of indices), so we will call it bidirectional. If the field is finite, a sequence is periodic if and only if it admits a bidirectional linear recurrence relation. For solving the above problem we propose an algorithm similar to the Berlekamp-Massey algorithm and prove its correctness. We also describe the set of all solutions to this problem and prove some properties of the minimal polynomials of initial segments of the sequence.

This dissertation is a study on Generating Functions and the Linear Recurrence Relations and is divided in three chapters in the following manner:On the first chapter we present Ordinary Generating Functions and Exponential Generating Functions along with their definitions, properties and a few theorems with their respective demonstrations.Still on this chapter we explore the concepts of Partitions and some Partition Generating Functions.On the second chapter we explore the concept of Recurrence Relations, specifically the Linear ones.We show a few methods for the resolution of Recurrence Relations, highlighting the resolution based on Generating Functions and also an interesting method using matrix diagonalization.The methods are very well detailed and are easy to comprehend.Chapter 3 is made of four applications on Recurrence Relations and Generating Functions.(n,k)-block fountain is a method of piling up coins on a plane obeying a few criteria.Steiner's pizza consists of dividing the pizza and obtaining the greatest number of slices with n cuts, triangulation of a convex n-agon is a problem of diving polygons in triangles and lastly one of the most famous sequences in Mathematics, the Fibonacci Sequence, where we show the sequence recurrence relation and its respective generating function.

Summary We describe a general method that finds closed forms for partial sums of power series whose coefficients arise from linear recurrence relations. These closed forms allow one to derive a vast collection of identities involving the Fibonacci numbers and other related sequences. Although motivated by a polynomial long division problem, the method fits naturally into a standard generating function framework. We also describe an explicit way to calculate the generating function of the Hadamard product of two generating functions, a construction on power series which resembles the dot product. This allows one to use the method for many examples where the recurrence relation for the coefficients is not initially known.