Related papers: Zeros Transfer For Recursively defined Polynomials

We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the…

In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial $f$, either $|f(A,B)|=\Omega(n^{4/3})$, for every pair of…

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…

The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in $n+m$ variables, which reduce to the Jack polynomials when $n=0$ or $m=0$ and provide joint eigenfunctions of the quantum integrals of the…

In this paper, we shall present an interesting and significant refinement of a classical result of Cauchy about the moduli of the zeros of a quaternionic polynomial. As an application of this result we shall obtain zero-free regions of…

We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…

This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as…

We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any…

We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra…

We present new polynomial-based methods for discrete-time quaternionic systems, highlighting how noncommutative multiplication modifies classical control approaches. Defining quaternionic polynomials via a backward-shift operator, we…

For a univariate real polynomial without zero coefficients, Descartes' rule of signs (completed by an observation of Fourier) says that its numbers $pos$ of positive and $neg$ of negative roots (counted with multiplicity) are majorized…

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…

We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg-Kohn functional for electronic or nucleonic densities and for the study…

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the…

For the generalized Jacobi, Laguerre and Hermite polynomials $P_n^{(\alpha_n, \beta_n)} (x), L_n^{(\alpha_n)} (x),$\break $H_n^{(\gamma_n)} (x)$ the limit distributions of the zeros are found, when the sequences $\alpha_n$ or $\beta_n$ tend…

We study equidistribution problem of zeros in relation to a sequence of $Z$-asymptotically Chebyshev polynomials on $\mathbb{C}^{m}$. We use certain results obtained in a very recent work of Bayraktar, Bloom and Levenberg and have an…

In this paper, we apply the ratio conjecture of $L$-functions to derive the lower order terms of the $1$-level density of the low-lying zeros of a family quadratic Hecke $L$-functions in the Gaussian field. Up to the first lower order term,…

The complex or non-hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the…

Hayes equivalence is defined on monic polynomials over a finite field $\fq$ in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial $Q$. We study the distribution of the number of zeros in a…

The paired Hayman's conjecture of different types are considered. More accurately speaking, the zeros of a pair of $f^nL(z,g)-a_1(z)$ and $g^mL(z,f)-a_2(z)$ are characterized using different methods from those previously employed, where $f$…