Ratio asymptotics of Hermite-Padé polynomials for Nikishin systems

Abstract

We prove the existence of ratio asymptotic for a sequence of multiple orthogonal polynomials which share orthogonality relations with a collection of m finite Borel measures supported on a bounded interval of the real line and constitute a so called Nikishin system of measures. When m = 1 our result reduces to E. A. Rakhmanov's known Theorem on ratio asymptotic for orthogonal polynomials on a segment.

Computation of conformal representations of compact Riemann surfaces

“…the functions ψ (1) and ψ (2) can be computed explicitly if we know α and a. As follows from the proof, the rational function G, given by this theorem, is given alternatively by (1.2), where g is one of the conformal homeomorphisms of C onto R. As part of the proof of Theorem 3.1 we will also obtain that G and the real numbers β, α, a, b constitute a unique solution of the following system of relations:…”

“…This is known as the Rakhmanov-Denisov theorem (see [3] and [9]). Recently (see [1], [2], and [5]), results analogous to those stated above and the Denisov-Rakhmanov theorem were obtained for multiple orthogonal polynomials of Nikishin systems of m measures. In this case, the ratio asymptotic is described in terms of a conformal representation of an (m+1)-sheeted compact Riemann surface onto the extended complex plane.…”

“…(1) |R (1) . In view of our motivation coming from the ratio asymptotic of multiple orthogonal polynomials of Nikishin systems of measures, we would like to find an explicit expression for ψ (l) or, at least, an algebraic equation characterizing it, in terms of the end points of the intervals ∆ k .…”

“…Their proofs are carried out in Sections 4 and 5. The final Section 6 is devoted to the development of a numerical algorithm for solving the system (Syst), which in turn allows one to calculate the functions ψ (1) , ψ (2) numerically (for m = 2).…”

Selfadjoint Jacobi matrices on graphs and multiple orthogonal polynomials

“…И если рекурренции вдоль диагональных "лест-ничных" линий ("step" lines) связывались с несимметричными разностными операторами высокого порядка (см. [17,9,10,11,18,19]), то соотношения (1.5), (1.7) для мультииндексов ⃗ := ( 1 , . .…”