Related papers: On Hyperbolic Polynomials and Four-term Recurrence…

For any fixed positive integer $n$, we study the root distribution of a sequence of polynomials $H_{m}(z)$ satisfying the rational generating function \[ \sum_{m=0}^{\infty}H_{m}(z)t^{m}=\frac{1}{1+B(z)t+A(z)t^{n}} \] where $A(z)$ and…

Let $ (G_n)_{n=0}^{\infty} $ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation $ G_n = f_1\alpha_1^n + \cdots + f_k\alpha_k^n $ and polynomial characteristic roots $…

We consider orthogonal polynomials $\{p_{n,N}(x)\}_{n=0}^{\infty}$ on the real line with respect to a weight $w(x)=e^{-NV(x)}$ and in particular the asymptotic behaviour of the coefficients $a_{n,N}$ and $b_{n,N}$ in the three term…

Our goal is to find an asymptotic behavior as $n\to\infty$ of orthogonal polynomials $P_{n}(z)$ defined by the Jacobi recurrence coefficients $a_{n}, b_{n}$. We suppose that the off-diagonal coefficients $a_{n}$ grow so rapidly that the…

Our interest lies in describing the zero behaviour of Gauss hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are arbitrary parameters. In general, this problem has not been solved and even when $b$ and $c$ are both real, the…

For any two $n$-$th$ order polynomials $a(s)$ and $b(s),$ the Hurwitz stability of their convex combination is necessary and sufficient for the existence of a polynomial $c(s)$ such that $c(s)/a(s)$ and $c(s)/b(s)$ are both strictly…

In this paper we consider interlacing of the zeros of polynomials from different sequences $\{p_n\}$ and $\{g_n\}$. In our main result we consider a mixed recurrence equation necessary for existence of a linear term $(x-A)$ so that the…

For each $\alpha>0$ and $A(z),B(z)\in\mathbb{C}[z]$, we study the zero distribution of the sequence of polynomials $\left\{ P_{m}^{(\alpha)}(z)\right\} _{m=0}^{\infty}$ generated by $(1+B(z)t+A(z)t^{3})^{-\alpha}$. We show that for large…

For any two n-th order polynomials a(s) and b(s), the Hurwitz stability of their convex combination is necessary and sufficient for the existence of a polynomial c(s) such that c(s)/a(s) and c(s)/b(s) are both strictly positive real.

For $0<\alpha<1$, we study the zeros of the sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $(1-t)^{\alpha}(1-2zt+t^{2})$, expanded as a power series in $t$. Equivalently, this sequence is…

Let $E\subset \mathbb Z$ be a set of positive upper density. Suppose that $P_1,P_2,..., P_k\in \mathbb Z[X]$ are polynomials having zero constant terms. We show that the set $E\cap (E-P_1(p-1))\cap ... \cap (E-P_k(p-1))$ is non-empty for…

Let $(P_n(x;z;\lambda))_{n\geq 0}$ be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional $\mathbf{s}$ defined by $$\langle\mathbf{s},p\rangle=\int_{-1}^1 p(x)(1-x^2)^{(\lambda-1/2)}…

We prove that there is an absolute constant $c > 0$ such that every polynomial $P$ of the form $$P(z) = \sum_{j=0}^{n}{a_jz^j}\,, \quad |a_0| = 1\,, \quad |a_j| \leq M\,, \quad a_j \in \Bbb{C}\,, \quad M \geq 1\,,$$ has at most…

Consider $\{p_n\}_{n=0}^{\infty}$, a sequence of polynomials orthogonal with respect to $w(x)>0$ on $(a,b)$, and polynomials $\{g_{n,k}\}_{n=0}^{\infty},k \in \mathbb{N}_0$, orthogonal with respect to $c_k(x)w(x)>0$ on $(a,b)$, where…

We give a characterization of common zeros of a sequence of univariate polynomials $W_n(z)$ defined by a recurrence of order two with polynomial coefficients, and with $W_0(z)=1$. Real common zeros for such polynomials with real…

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables . Assume that this polynomial satisfies the property : \\ $|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) :…

Given $H\subseteq \mathbb{C}$ two natural objects to study are the set of zeros of polynomials with coefficients in $H$, $$\{z\in \mathbb{C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum_{n=0}^{k}a_{n}z^n=0\},$$ and the set of zeros of…

Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017,…

A positive linear recurrence sequence is of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n+1-L}$ with each $c_i \ge 0$ and $c_1 c_L > 0$, with appropriately chosen initial conditions. There is a notion of a legal decomposition (roughly,…

Given that $a,b\in\mathbb N$, $c_0,c_1\in\mathbb Z$, $(c_0,c_1)\neq (0,0)$, and a generalized Fibonacci sequence $(s_n)_{n\geq 0}$ where $s_0 = c_0$, $s_1 = c_1$, and $s_{n+1}=as_{n}+bs_{n-1}$ for all positive integers $n$. In this paper,…