Abstract:
The geometry of polynomials explores geometrical relationships between the zeros and the coefficients of a polynomial. A classical problem in this theory is to locate the zeros of a given polynomial by determining disks in the complex plane in which all its zeros are situated. In this paper, we infer bounds for general polynomials and apply classical and new results to graph polynomials namely Wiener and distance polynomials whose zeros have not been yet investigated. Also, we examine the quality of such bounds by considering four graph classes and interpret the results.

Abstract:
The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A generalization of the classical Heine-Mehler formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros. We also describe the location and the asymptotic behaviour of the exceptional zeros, which converge for large n to fixed values.

Abstract:
We derive upper bounds for the smallest zero and lower bounds for the largest zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed three term recurrence relations satisfied by polynomials corresponding to different parameter(s) within the same classical family. We prove that interlacing properties of the zeros impose restrictions on the possible location of common zeros of the polynomials involved and deduce strict bounds for the extreme zeros of polynomials belonging to each of these three classical families. We show numerically that the bounds generated by our method improve known lower (upper) bounds for the largest (smallest) zeros of polynomials in these families, notably in the case of Jacobi and Gegenbauer polynomials.

Abstract:
We investigate the zeros of a family of hypergeometric polynomials $_2F_1(-n,-x;a;t)$, $n\in\nn$ that are known as the Meixner polynomials for certain values of the parameters $a$ and $t$. When $a=-N$, $N\in\nn$ and $t=\frac1{p}$, the polynomials $K_n(x;p,N)=(-N)_n\phantom{}_2F_1(-n,-x;-N;\frac1{p})$, $n=0,1,...N$, $0

Abstract:
Let $f$ and $F$ be two polynomials satisfying $F(x)=u(x)f(x)+v(x)f'(x)$. We characterize the relation between the location and multiplicity of the real zeros of $f$ and $F$, which generalizes and unifies many known results, including the results of Brenti and Br\"and\'en about the $q$-Eulerian polynomials.

Abstract:
We study the zeros of exceptional Hermite polynomials associated with an even partition $\lambda$. We prove several conjectures regarding the asymptotic behavior of both the regular (real) and the exceptional (complex) zeros. The real zeros are distributed as the zeros of usual Hermite polynomials and, after contracting by a factor $\sqrt{2n}$, we prove that they follow the semi-circle law. The non-real zeros tend to the zeros of the generalized Hermite polynomial $H_{\lambda}$, provided that these zeros are simple. It was conjectured by Veselov that the zeros of generalized Hermite polynomials are always simple, except possibly for the zero at the origin, but this conjecture remains open.

Abstract:
It is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five-diagonal matrices depending on the Schur parameters. This result is achieved through the study of orthogonal Laurent polynomials on the unit circle. More precisely, it is a consequence of the five term recurrence relation obtained for these orthogonal Laurent polynomials, and the one to one correspondence established between them and the orthogonal polynomials on the unit circle. As an application, some results relating the behavior of the zeros of orthogonal polynomials and the location of Schur parameters are obtained.

Abstract:
We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.

Abstract:
We find approximate expressions and for the real and imaginary parts of the th zero of the Bessel polynomial . To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions of , and is obtained. It is shown that the resulting complex number is -convergent to for fixed . 1. Introduction The polynomial solutions of the differential equation were studied systematically in [1] by the first time. They are named (generalized) Bessel polynomials and are given explicitly by as it can be shown in [2]. Here, is the Pochhammer symbol and . Many properties as well as applications are associated to this equation; the traveling waves in the radial direction which are solutions of the wave equation in spherical coordinates can be written in terms of the polynomial solutions of (1.1). Also, this equation has application in network and filter design, isotropic turbulence fields, and more (see the monograph [2] or [3–14] and references therein for some other results). Among these, several results about the important problem concerning the location of its zeros have been obtained [8–11] and in [12], explicit expressions for sum rules and for the homogeneous product sum symmetric functions of zeros of these polynomials are given. On the other hand, the electrostatic interpretation of these zeros as the equilibrium configuration in the complex plane with a logarithmic electric potential and a dipole at the origin has been given in [13], and in [14] it is shown that this equilibrium configuration is not stable. Thus, these cases show that it is desirable to acquire new analytical knowledge about the location of the zeros of the Bessel polynomials. In this paper we give approximate explicit formulas for both the real and imaginary parts of the th zero of and show that the approximation order of these new formulas to the exact zeros of the Bessel polynomials is for fixed . The approach followed in this paper is simple and based on three items. The first is the electrostatic interpretation of the zeros of polynomials satisfying second-order differential equations [15–17], the second is a simple curve fitting of numerical data, and the third is the known fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials