Related papers: Determinantal and permanental representation of ge…
In this paper, we give some determinantal and permanental representations of Generalized Fibonacci Polynomials by using various Hessenberg matrices. These results are general form of determinantal and permanental representations of k…
In this paper, we give some determinantal and permanental representations of Generalized Lucas Polynomials by using various Hessenberg matrices, which are general form of determinantal and permanental representations of ordinary Lucas and…
We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…
In this paper we construct two types of Hessenberg matrices with the properties that every weighted isobaric polynomial (WIP) appears as a determinant of one of them, and as the permanent of the other. Every integer sequence which is…
A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials,…
In this note, we obtain some identities for the generalized Fibonacci polynomial by using the Q(x) matrix. These identities including the Cassini identity and Honsberger formula can be applied to some polynomial sequences, such as Fibonacci…
In this article we study a generalization of Fibonomials, replacing the Fibonacci sequences by bivariate s-Fibonacci polynomial sequences. We call the obtained objects "Bivariate s-Fibopolynomials".
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
In this study, the new algebraic properties related to bivariate Fibonacci polynomials has been given. We present the partial derivatives of these polynomials in the form of convolution of bivariate Fibonacci polynomials. Also, we define a…
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…
In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…
Determinantal polynomials play a crucial role in semidefinite programming problems. Helton-Vinnikov proved that real zero (RZ) bivariate polynomials are determinantal. However, it leads to a challenging problem to compute such a…
The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s and t given by {0} = 0, {1} = 1, and {n} = s{n-1}+t{n-2} for n ge 2. The latter are defined…
A second order polynomial sequence is of Fibonacci type (Lucas type) if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers. In this paper we generalize identities from Fibonacci numbers and…
Starting with some determinants of binomial coefficients which are related to Fibonacci and Lucas polynomials we study similar determinants for some generalizations of these polynomials and their q-analogues.
We consider two type of upper Hessenberg matrices which determinants are Fibonacci numbers. Calculating sums of principal minors of the fixed order of the first type leads us to convolved Fibonacci numbers. Some identities for these and for…
In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers…
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 prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation of hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove…
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.