Dynamics of the Zeros of Fibonacci Polynomials

Zeros of a Class of Fibonacci-type Polynomials

with the initial values G0(x) = a and G1(x) = x + b. The polynomials Gn−1(1, 0;x) and Gn(2, 0;x)(n ≥ 1) are just the usual Fibonacci polynomials Fn(x) and the Lucas polynomials Ln(x), respectively. Our concern in this paper is to study some of the properties of the zeros of the Fibonacci-type polynomials Gn(a, b;x). The zeros of Fn(x) and Ln(x) have been given explicitly by V. E. Hoggatt, Jr. a...

Cube polynomial of Fibonacci and Lucas cubes

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...

Some compact generalization of inequalities for polynomials with prescribed zeros

‎Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$‎, ‎$k^2 leq rRleq R^2$ and for $Rleq r leq k$‎. ‎Our results refine and generalize certain well-known polynomial inequalities‎.

Domain of attraction of normal law and zeros of random polynomials

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...

On the average number of sharp crossings of certain Gaussian random polynomials