Jacobsthal Polynomials Research Articles

Binomial Transforms of the Third-Order Jacobsthal and Modified Third-Order Jacobsthal Polynomials

In this study, we define the binomial transforms of third-order Jacobsthal and modified third-order Jacobsthal polynomials. Further, the generating functions, Binet formulas and summation of these binomial transforms are found by recurrence relations. Also, we establish the relations between these transforms by deriving new formulas. Finally, the Vajda, d'Ocagne, Catalan and Cassini formulas for these transforms are obtained.

Some properties of combinatorial triangles related to Horadam polynomials

The Horadam polynomials unify many well-known polynomials, such as the Fibonacci polynomials, the Lucas polynomials, the Pell polynomials, the Jacobsthal polynomials and the Chebyshev polynomials. We investigate some properties of various combinatorial triangles related to Horadam polynomials, including their properties as almost-Riordan arrays and Riordan arrays, their total positivity, the real-rootedness of the generating functions of their rows, and the asymptotic normality (by central and local limit theorems).

Infinite Sums Involving Jacobsthal Polynomials: Generalizations

Infinite Sums Involving Jacobsthal Polynomials: Generalizations

P-Numerical Semigroups of Generalized Fibonacci Triples

For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup Sp is defined as the set of integers whose nonnegative integral linear combinations of given positive integers a1,a2,…,ak are expressed in more than p ways. When p=0, S0 with the 0-Frobenius number and the 0-genus is the original numerical semigroup with the Frobenius number and the genus. In this paper, we consider the p-numerical semigroup involving Jacobsthal polynomials, which include Fibonacci numbers as special cases. We can also deal with the Jacobsthal–Lucas polynomials, including Lucas numbers accordingly. An application on the p-Hilbert series is also provided. There are some interesting connections between Frobenius numbers and geometric and algebraic structures that exhibit symmetry properties.

Sums Involving Jacobsthal Polynomials

Sums Involving Jacobsthal Polynomials

Infinite Sums Involving Jacobsthal Polynomials Revisited

Infinite Sums Involving Jacobsthal Polynomials Revisited

Infinite Sums Involving Jacobsthal Polynomials

Infinite Sums Involving Jacobsthal Polynomials

Products Involving Reciprocals of Jacobsthal Polynomials

Products Involving Reciprocals of Jacobsthal Polynomials

On New Polynomial Sequences Constructed to Each Vertex in an n‐Gon

In this work, we bring to light the properties of newly formed polynomial sequences at each vertex of Pell polynomial sequences placed clockwise at each vertex in the n‐gon. We compute the relation among the polynomials with such vertices. Moreover, in an n‐gon, we generate a recurrence relation for a sequence giving the mth term formed at the kth. Similar to the situations we are talking about, we applied to Pell‐Lucas polynomials, Jacobsthal polynomials, and Jacobsthal‐Lucas polynomials and have obtained new relations recurrence.

Catalan transform of the incomplete Jacobsthal numbers and incomplete generalized Jacobsthal polynomials

We introduce the Catalan transform of the Incomplete Jacobsthal numbers. We apply the Hankel transform to the Catalan transforms of these numbers. We calculate determinants of matrixes formed with [Formula: see text]by using Hankel transform. Then we define the incomplete h(x)-Jacobsthal polynomials. Then we examine the recurrence relation and some properties of these polynomials.

Some identities of generalized Tribonacci and Jacobsthal polynomials

In this study, we denote $(t'_{n}(x))_{n\in \mathbb{N}}$ the generalized Tribonacci polynomials, which are defined by $t'_{n}(x)=x^{2}t'_{n-1}(x)+xt'_{n-2}(x)+t'_{n-3}(x), n \geqslant 4,$ with $t_{1}(x)=a, t_{2}(x)=b, t_{3}(x)=cx^{2}$ and we drive an explicit formula of $(t'_{n}(x))_{n\in \mathbb{N}}$ in terms of their coefficients $T'(n,j)$, Also, we establish some properties of $(t_{n}(x))_{n\in \mathbb{N}}$. Similarly, we study the Jacobsthal polynomials $(J_{n}(x))_{n\in \mathbb{N}}$, where $J_{n}(x)=J_{n-1}(x)+x J_{n-2}(x)+ x^{2} J_{n-3}(x), n \geqslant 4$, with $J_{1}(x)= J_{2}(x)=1, J_{3}(x)=x+1$ and describe some properties.

On third-order Jacobsthal polynomials and their properties

On third-order Jacobsthal polynomials and their properties

A new class of generating functions of binary products of Gaussian numbers and polynomials

In this paper, we introduce a operator in order to derive some new symmetric properties of Gaussian Fibonacci numbers and Gaussian Lucas numbers. By making use of the operator defined in this paper, we give some new generating functions for Gaussian Fibonacci numbers and Gaussian Jacobsthal polynomials. In the paper Al4, Al5, a second-order linear recurrence sequence (U_{n}(a,b;p,q))_{n≥0} or briefly (U_{n})_{n≥0} is considered by the recurrence relation: U_{n+2}=pU_{n+1}+qU_{n}, with the initial conditions U₀=a and U₁=b, where a,b∈ℂ and p,q∈ℤ₊ for n≥0.

A NEW FAMILIES OF GAUSS k-JACOBSTHAL NUMBERS AND GAUSS k-JACOBSTHAL-LUCAS NUMBERS AND THEIR POLYNOMIALS

In this paper, we define the new families of Gauss k-Jacobsthal numbers and Gauss k-Jacobsthal-Lucas numbers. We obtain some exciting properties of the families. We give the relationships between the family of the Gauss k-Jacobsthal numbers and the known Gauss Jacobsthal numbers, the family of the Gauss k-Jacobsthal-Lucas numbers and the known Gauss Jacobsthal-Lucas numbers. We also define the generalized polynomials for these numbers. Further, we obtain some interesting properties of the polynomials. In addition, we give the relationships between the generalized Gauss k-Jacobsthal polynomials and the known Gauss Jacobsthal polynomials, the generalized Gauss k-Jacobsthal-Lucas polynomials and the known Gauss Jacobsthal-Lucas polynomials. Furthermore, we find the new generalizations of these families and the polynomials in matrix representation. Then we prove Cassini’s Identities for the families and their polynomials.

On the sum of reciprocals of Jacobsthal polynomials

In this paper, we have obtained the infinite sum obtained from the reciprocals of Jacobsthal polynomial and Jacobsthal-Lucas polynomial. Also, we have obtained some inequalities involving the infinite sum of reciprocals of squares of Jacobsthal polynomial and Jacobsthal-Lucas polynomial.

On Generalized Jacobsthal and Jacobsthal-Lucas polynomials

Abstract In this paper we introduce a generalized Jacobsthal and Jacobsthal-Lucas polynomials, Jh,n and jh,n, respectively, that consist on an extension of Jacobsthal's polynomials Jn(𝑥) and Jacobsthal-Lucas polynomials jn(𝑥). We provide their properties and a generalization of the usual identities. We also present, for each one of these generalized polynomials, their ordinary generating functions and matrices. In the last part of the paper, we present some special kind of tridiagonal matrices whose entries are elements of these generalized polynomials.

Determinantal and permanental representations of Fibonacci type numbers and polynomials

In this paper, we compute terms of the matrix $A_{(k)}^{\infty }$, which contains Fibonacci type numbers and polynomials, with the help of determinants and permanents of various Hessenberg matrices. In addition, we show that determinants of these Hessenberg matrices can be obtained by using combinations. The results that we obtain are important, since the matrix $A_{(k)}^{\infty } $ is a general form of Fibonacci type numbers and polynomials, such as $k$ sequences of the generalized order-$k$ Fibonacci and Pell numbers, generalized bivariate Fibonacci $p$-polynomials, bivariate Fibonacci and Pell $p$-polynomials, second kind Chebyshev polynomials and bivariate Jacobsthal polynomials, etc.

Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers

The purpose of this paper is to construct generating functions for the family of the Fibonacci and Jacobsthal polynomials. Using these generating functions and their functional equations, we investigate some properties of these polynomials. We also give relationships between the Fibonacci, Jacobsthal, Chebyshev polynomials and the other well known polynomials. Finally, we give some infinite series applications related to these polynomials and their generating functions.

Generalized Jacobsthal polynomials and special points for them

In this work we introduce a family of polynomials that satisfy the recurrence relations for Jacobsthal polynomials with generalized initial conditions by analogy to work of V.K. Gupta, Y.K. Panwar, and O. Sikhwal from 2012. Explicit closed form and the Binet formulas for the generalized Jacobsthal polynomials are presented. The generating function and other relations for them are also found. Special points for this family are analyzed and presented pictorially.

Convolutions of the generalized Morgan–Voyce polynomials

We consider the following classes of polynomials: B n , m ( s ) ( x ) , C n , m ( s ) ( x ) , b n , m ( s ) ( x ) and c n , m ( s ) ( x ) , which are convolutions of the generalized Morgan-Voyce polynomials, where s is a nonnegative integer. These convolutions are related to the classical Morgan-Voyce, Chebyshev and Jacobsthal polynomials.