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 satisfying remarkable recurrence relations.

In this paper we generalize to bivariate Fibonacci and Lucas polynomials, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate bases are families of integers satisfying remarkable recurrence relations.

The Fibonacci polynomials are defined by the recursion relation Fn+2{x) = xF„+l(x) + Fn(x), (1) with the initial values Fx(x) = 1 and F2(x) = x. When x = l, Fn(x) is equal to the /1 Fibonacci number, Fn. The Lucas polynomials, Ln(x) obey the same recursion relation, but have initial values Li(x) = x and L^x) = x +2. Explicit expressions for the zeros of the Fibonacci and Lucas polynomials have ...

The Hosoya polynomial triangle is a triangular arrangement of polynomials where each entry is a product of two polynomials. The geometry of this triangle is a good 1 tool to study the algebraic properties of polynomial products. In particular, we find closed formulas for the alternating sum of products of polynomials such as Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials...

Fibonacci cubes, extended Fibonacci cubes, and Lucas cubes are induced subgraphs of hypercubes 9 defined in terms of Fibonacci strings. It is shown that all these graphs are median. Several enumeration results on the number of their edges and squares are obtained. Some identities involving Fibonacci 11 and Lucas numbers are also presented. © 2005 Published by Elsevier B.V. 13

We study noncommutative continuant polynomials via a new leapfrog construction. This needs the introduction of new indeterminates and leads to generalizations of Fibonacci polynomials, Lucas polynomials and other families of polynomials. We relate these polynomials to various topics such as quiver algebras and tilings. Finally, we use permanents to give a broad perspective on the subject.

The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k ≥ 0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonac...

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.

Abstract Nonorthogonal polynomials have many useful properties like being used as a basis for spectral methods, generated in an easy way, having exponential rates of convergence, fewer terms and reducing computational errors comparison with some others, producing most important basic polynomials. In this regard, paper deals new indirect numerical method to solve fractional optimal control probl...