Related papers: Zeros of polynomials with four-term recurrence and…

For any real numbers $b,c\in\mathbb{R}$, we form the sequence of polynomials $\left\{ H_{m}(z)\right\} _{m=0}^{\infty}$ satisfying the four-term recurrence \[ H_{m}(z)+cH_{m-1}(z)+bH_{m-2}(z)+zH_{m-3}(z)=0,\qquad m\ge3, \] with the initial…

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…

For any $A(z),B(z),C(z)\in\mathbb{C}[z]$, we study the zero distribution of a table of polynomials $\left\{ P_{m,n}(z)\right\} _{m,n\in\mathbb{N}_{0}}$ satisfying the recurrence relation \[…

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…

From a sequence $\left\{ a_{n}\right\} _{n=0}^{\infty}$ of real numbers satisfying a three-term recurrence, we form a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ whose coefficients are numbers in this sequence. We…

This paper discusses the location of zeros of polynomials in a polynomial sequence $\{P_n(z)\}$ generated by a three-term recurrence relation of the form $P_n(z)+ B(z)P_{n-1}(z) +A(z) P_{n-k}(z)=0$ with $k>2$ and the standard initial…

For any real numbers $a,\ b$, and $c$, we form the sequence of polynomials $\{P_n(z)\}_{n=0}^\infty$ satisfying the four-term recurrence \[ P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in\mathbb{N}, \] with the initial conditions…

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions…

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence $\{P_i\}_{i=1}^{\infty}$ generated by a three-term recurrence relation $P_i(x)+ Q_1(x)P_{i-1}(x) +Q_2(x) P_{i-2}(x)=0$…

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 $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z)…

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…

We describe the limit zero distributions of sequences of polynomials with positive coefficients.

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a denominator of the form $G(z,t)=P(t)+zt^{r}$, where the zeros of $P$ are positive and real. We show that every member of…

In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the…

We study the root distribution of a sequence of polynomials $\{P_n(z)\}_{n=0}^{\infty}$ with the rational generating function $$ \sum_{n=0}^{\infty} P_n(z)t^n= \frac{1}{1+ B(z)t^\ell +A(z)t^k}$$ for $(k,\ell)=(3,2)$ and $(4,3)$ where $A(z)$…

We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly…

In this paper, we study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc,…

Our goal in this paper is to study the zero distribution of a sequence of polynomials whose coefficients satisfy a three-term recurrence. Equivalently, these polynomials are Taylor polynomials of a rational function with a polynomial…

We investigate the sequence $(P_{n}(z))_{n=0}^{\infty}$ of random polynomials generated by the three-term recurrence relation $P_{n+1}(z)=z P_{n}(z)-a_{n} P_{n-1}(z)$, $n\geq 1$, with initial conditions $P_{\ell}(z)=z^{\ell}$, $\ell=0, 1$,…