Related papers: Zero distribution of multiplicative Hermite and La…

In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by…

Let $x_1, \dots, x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots, x_n$. We…

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 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…

We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular…

The backward heat flow on the real line started from the initial condition $z^n$ results in the classical $n$-th Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large $n$ limit. Similarly, the…

We study Bessel and Dunkl processes $(X_{t,k})_{t\ge0}$ on $\mathbb R^N$ with possibly multivariate coupling constants $k\ge0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For…

We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degree $d$ trigonometric polynomial are distributed according to some probability…

We consider the problem of zero distribution of the first kind Hermite--Pad\'e polynomials associated with a vector function $\vec f = (f_1, \dots, f_s)$ whose components $f_k$ are functions with a finite number of branch points in plane.…

We study the asymptotic zero distribution of the rescaled Laguerre polynomials, $\displaystyle L_n^{(\alpha_n)}(nz)$, with the parameter $\alpha_n$ varying in such a way that $\displaystyle \lim_{n\rightarrow \infty}\alpha_n/n=-1$. The…

We study the probability that all the eigenvalues of $n\times n$ Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^{\gamma}\mathrm{e}^{-4nx},\;x\in[0,\infty),\;\gamma>-1$, lie in the interval $[0,\alpha]$. By using…

For a positive integer $n$ and a real number $\alpha$, the generalized Laguerre polynomials are defined by \begin{align*} L^{(\alpha)}_n(x)=\sum^n_{j=0}\frac{(n+\alpha)(n-1+\alpha)\cdots (j+1+\alpha)(-x)^j}{j!(n-j)!}. \end{align*} These…

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…

We prove expressions for the inequalities in Hermite's theorem which are conditions for a real polynomial to have real zeros. These expressions generalize the discriminant of a quadratic polynomial and the expression of J. Mar\'ik for a…

We consider random polynomials of the form $H_n(z)=\sum_{j=0}^n\xi_jq_j(z)$ where the $\{\xi_j\}$ are i.i.d non-degenerate complex random variables, and the $\{q_j(z)\}$ are orthonormal polynomials with respect to a compactly supported…

We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $N\times (N+\nu)$ are correlated through a non-Hermiticity parameter $\tau\in[0,1]$.…

Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to $r>1$ normal (Gaussian) weights $w_j(x)=e^{-x^2+c_jx}$ with different means $c_j/2$, $1 \leq j…

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\binom{j}{2} \omega+jy+x \mod 1$ for irrational $\omega$. We prove that the eigenvalue distribution of…

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…

A new characterization of the generalized Hermite polynomials and of the orthogonal polynomials with respect to the maesure $|x|^\g (1-x^2)^{\a-1/2}dx$ is derived which is based on a "reversing property" of the coefficients in the…