Related papers: On zeros of polynomials and allied functions satis…
The aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way, and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials we associate…
We study the derivatives of polynomials with equally spaced zeros and find connections to the values of the Riemann zeta-function at the positive even integers.
In this paper we investigate distribution of zeros for once quasipolynom and obtain exactly lower-bound for their modulus.
In this paper, we obtain several new classes of irreducible polynomials having integer coefficients whose zeros lie inside an open disk around the origin or outside a closed annular region in the complex plane. Such irreducible polynomials…
We give the asymptotic distribution of the zeros of Jacobi-Pi\~neiro polynomials and multiple Laguerre polynomials of the first kind. We use the nearest neighbor recurrence relations for these polynomials and a recent result on the ratio…
Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to…
We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where \lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n…
The existence of the limit distribution of the zeros of Hermite-Pad\'e polynomials of type II for a pair of functions forming a Nikishin system is proved using the scalar equilibrium problem posed on the two-sheeted Riemann surface. The…
The zeros of D'Arcais polynomials, also known as Nekrasov--Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind functions. These polynomials satisfy difference equations of hereditary type with…
In this paper, we provide a complete Plancherel-Rotach asymptotic analysis of polynomials that satisfy a second-order difference equation with linear coefficients. According to the signs of the parameters, we classify the difference…
The $X_1$-Jacobi and the $X_1$-Laguerre exceptional orthogonal polynomials have been introduced and studied by G\'omez-Ullate, Kamran and Milson in a series of papers. In this note, we establish some properties, such as interlacing,…
In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.
We present sharp lower bounds for the A-numerical radius of semi-Hilbertian space operators. We also present an upper bound. Further we compute new upper bounds for the $B$-numerical radius of $2 \times 2$ operator matrices where $B =…
We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of…
Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if…
For random polynomials with i.i.d. (independent and identically distribu-ted) zeros following any common probability distribution $\mu$ with support contained in the unit circle, the empirical measures of the zeros of their first and higher…
The main aim of this work is to apply the matrix approach of ortho\-gonal polynomials associated with infinite Hermitian definite positive matrices in relation with an important question regarding the location of zeros of Sobolev orthogonal…
In this paper, we study the value distribution of zeros of certain nonlinear difference polynomials of entire functions of finite order.
A method for finding relations for the roots of polynomials is presented. Our approach allows us to get a number of relations for the zeros of the classical polynomials and for the roots of special polynomials associated with rational…
In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this…