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Evgenii A. Rakhmanov

Publications and source records attributed to Evgenii A. Rakhmanov.

5 recordsLinked to original sources
arXiv2025

Flow of the zeros of polynomials under iterated differentiation

For a monic polynomial $Q_n$ of degree $n$, let $Q_{n, k}$ be its $k$-th derivative normalized to be monic. Under the only assumption that the sequence $\{Q_n\}$ has a weak* limiting zero distribution (an empirical distribution of zeros) represented by a probability measure $μ_0$ with compact support in the complex plane, we show that as $n, k \rightarrow \infty$ such that $k / n \rightarrow t \in(0,1)$, the Cauchy transform of the normalized zero-counting measure of the polynomials $Q_{n, k}$ converges in a neighborhood of infinity to an analytic function, uniquely determined by $μ_0$ and $t$, that can be written as the Cauchy transform of a measure $μ_t$, not necessarily uniquely determined unless $μ_0$ is supported on the real line. The family of these Cauchy transforms and, when well defined, the corresponding measures $μ_t $, $t \in(0,1)$, whose dependence on the parameter $t$ can be interpreted as a flow of the zeros under iterated differentiation, has several interesting connections with the inviscid Burgers equation, the fractional free convolution of $μ_0$, or a nonlocal diffusion equation governing the density of $μ_t$ on $\mathbb R$. We provide an elementary and unified approach that not only recovers, but also explains various phenomena observed in prior works - from Burgers-type PDEs to free probability limits.

math.CA↗
arXiv2025

Weighted equilibrium and the flow of derivatives of polynomials

Given a sequence of polynomials $Q_n$ of degree $n$ with zeros on $[-1,1]$, we consider the triangular table of derivatives $Q_{n, k}(x)=d^k Q_n(x) /d x^k$. Under the assumption that the sequence $\{Q_n\}$ has a weak* limiting zero distribution (an empirical distribution of zeros) given by the arcsine law, we show that as $n, k \rightarrow \infty$ such that $k / n \rightarrow t \in[0,1)$, the zero-counting measure of the polynomials $Q_{n, k}$ converges to an explicitly given measure $μ_t$. This measure is the equilibrium measure of $[-1,1]$ of size $1-t$ in an external field given by two mass points of size $t/2$ fixed at $\pm 1$. The main goal of this paper is to provide a direct potential theory proof of this fact.

math.CA↗
arXiv2015

Asymptotics of type I Hermite-Padé polynomials for semiclassical functions

Type I Hermite--Padé polynomials for a set of functions $f_0, f_1, ..., f_s$ at infinity, $Q_{n,0}$, $Q_{n,1}$, ..., $Q_{n,s}$, is defined by the asymptotic condition $$ R_n(z):=\bigl(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s\bigr)(z) =\mathcal O (\frac1{z^{s n+s}}), \quad z\to\infty, $$ with the degree of all $Q_{n,k}\leq n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\to \infty$ under the assumption that all $f_j$'s are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients. We discuss in more detail the case when $f_k$'s are powers of the same function $f$ ($f_k=f^k$); for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed. From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Padé approximants (when $s=1$).

math.CA↗
arXiv2011

Heine, Hilbert, Pade, Riemann, and Stieltjes: a John Nuttall's work 25 years later

In 1986 J. Nuttall published in Constructive Approximation the paper "Asymptotics of generalized Jacobi polynomials", where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Pade approximants to a special class of algebraic functions with 3 branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this paper features also a special appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters.

math.CA↗