High order recurrence relation, Hermite-Pad\'e approximation, and Nikishin systems

TL;DR

This paper explores the asymptotic behavior and orthogonality properties of polynomial sequences satisfying high order recurrence relations, linking them to Hermite-Padé approximation and Nikishin systems, especially for constant coefficient cases.

Contribution

It establishes new connections between high order recurrence relations, multiple orthogonal polynomials, and Nikishin systems, extending known results for Chebyshev polynomials.

Findings

Polynomials satisfy multiple orthogonality relations with Nikishin systems.

Results match known behaviors for Chebyshev polynomials.

Provides insights into eigenvalue distribution of banded Toeplitz matrices.

Abstract

The study of sequences of polynomials satisfying high order recurrence relations is connected with the asymptotic behavior of multiple orthogonal polynomials, the convergence properties of type II Hermite-Pad\'e approximation, and eigenvalue distribution of banded Toeplitz matrices. We present some results for the case of recurrences with constant coefficients which match what is known for the Chebyshev polynomials of the first kind. In particular, under appropriate assumptions, we show that the sequence of polynomials satisfies multiple orthogonality relations with respect to a Nikishin system of measures.