Alexander Ivanovich Aptekarev

Multiple orthogonal polynomials for classical weights

A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to p > 1 weights satisfying Pearsons equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order p + 1. We also obtain explicit formulas and recurrence relations for these polynomials.

Scalar and matrix Riemann-Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Pade approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann-Hilbert problem, are presented. The first method uses a scalar Riemann-Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Rieman-Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.

Strong asymptotics of Laguerre polynomials and information entropies of two-dimensional harmonic oscillator and one-dimensional Coulomb potentials

The information entropies of the two-dimensional harmonic oscillator, V(x,y)=1/2λ(x2+y2), and the one-dimensional hydrogen atom, V(x)=−1/|x|, can be expressed by means of some entropy integrals of Laguerre polynomials whose values have not yet been analytically determined. Here, we first study the asymptotical behavior of these integrals in detail by extensive use of strong asymptotics of Laguerre polynomials. Then, this result (which is also important by itself in a context of both approximation theory and potential theory) is employed to analyze the information entropies of the aforementioned quantum-mechanical potentials for the very excited states in both position and momentum spaces. It is observed, in particular, that the sum of position and momentum entropies has a logarithmic growth with respect to the main quantum number which characterizes the corresponding physical state. Finally, the rate of convergence of the entropies is numerically examined.

Asymptotics of Hermite-Padé Polynomials

We review results about the asymptotic behavior (in the strong and weak sense) of Hermite-Pade polynomials of type II (also known as German polynomials). The polynomials appear as numerators and denominators of simultaneous rational approximants. The survey begins with general remarks on Hermite-Pade polynomials and a short summary of the state of the theory in this field.

Asymptotics of Hermite–Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0)

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.

Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation

Transformations of the measure of orthogonality for orthogonal polynomials, namely Freud transformations, are considered. Jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations. Dynamics of the coefficients are described by generalized Toda equations. The classical Toda lattice equations are the simplest special case of dynamics of the coefficients under the Freud transformation of the measure of orthogonality. Also dynamics of Hankel determinants, its minors and other notions corresponding to the orthogonal polynomials are studied.

Asymptotics of orthogonal polynomial's entropy

This is a brief account on some results and methods of the asymptotic theory dealing with the entropy of orthogonal polynomials for large degree. This study is motivated primarily by quantum mechanics, where the wave functions and the densities of the states of solvable quantum-mechanical systems are expressed by means of orthogonal polynomials. Moreover, the uncertainty principle, lying in the ground of quantum mechanics, is best formulated by means of position and momentum entropies. In this sense, the behavior for large values of the degree is intimately connected with the information characteristics of high energy states. But the entropy functionals and their behavior have an independent interest for the theory of orthogonal polynomials. We describe some results obtained in the last 15 years, as well as sketch the ideas behind their proofs.

Criterion for the resolvent set of nonsymmetric tridiagonal operators

We study nonsymmetric tridiagonal operators acting in the Hilbert space ?2 and describe the spectrum and the resolvent set of such operators in terms of a continued fraction related to the resolvent. In this way we establish a connection between Pade approximants and spectral properties of nonsymmetric tridiagonal operators.

On the limit behavior of recurrence coefficients for multiple orthogonal polynomials

In this paper we investigate general properties of the coefficients in the recurrence relation satisfied by multiple orthogonal polynomials. The results include as particular cases Angelesco and Nikishin systems.

Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials

Let f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points,