Multiple orthogonal polynomials associated with an exponential cubic weight

TL;DR

This paper explores multiple orthogonal polynomials linked to an exponential cubic weight, analyzing their properties, recurrence relations, and asymptotic behaviors, revealing connections to discrete Painlevé equations.

Contribution

It introduces and studies a new class of multiple orthogonal polynomials with properties and asymptotics related to discrete Painlevé equations.

Findings

Recurrence coefficients satisfy a discrete Painlevé equation

Asymptotic behavior of recurrence coefficients characterized

Zeros and polynomial ratios analyzed

Abstract

We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and nearest-neighbor recurrence relations. It turns out that the recurrence coefficients are related to a discrete Painlev\'e equation. The asymptotics of the recurrence coefficients, the ratio of the diagonal multiple orthogonal polynomials and the (scaled) zeros of these polynomials are also investigated.