Computing recurrence coefficients of multiple orthogonal polynomials

TL;DR

This paper develops methods to compute recurrence coefficients of multiple orthogonal polynomials, linking step-line and nearest neighbor recurrence relations to facilitate calculations of these coefficients.

Contribution

It introduces algorithms to derive nearest neighbor recurrence coefficients from step-line coefficients and vice versa, advancing computational techniques for multiple orthogonal polynomials.

Findings

Derived formulas connecting step-line and nearest neighbor recurrence coefficients.

Provided algorithms for computing recurrence coefficients from each other.

Enhanced computational methods for multiple orthogonal polynomials.

Abstract

Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a $(r+2)$-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence relation) and there is a system of $r$ recurrence relations connecting the nearest neighbors (the so-called nearest neighbor recurrence relations). In this paper we deal with two problems. First we show how one can obtain the nearest neighbor recurrence coefficients (and in particular the recurrence coefficients of the orthogonal polynomials for each of the defining measures) from the step-line recurrence coefficients. Secondly we show how one can compute the step-line recurrence coefficients from the recurrence coefficients of the orthogonal polynomials of each of the measures defining the multiple orthogonality.