Computing recurrence coefficients of 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.

Discrete integrable systems generated by Hermite-Padé approximants

“…Remark 5.3. There is one more algorithm available, which is obtained by combining the well-known Jacobi-Perron algorithm (see [24] for the details) and a result from [12]. Namely, the Jacobi-Perron algorithm expands the vector (f 1 , f 2 ), where f 1 and f 2 form a perfect system, into a vector continued fraction.…”

On 2D discrete Schrödinger operators associated with multiple orthogonal polynomials

“…As one can see from the example constructed in Section 3, if the condition (5.3) is not satisfied then the operators H 1 , H 2 can not be uniquely determined. An algorithm for computing the coefficients a n,m , b n,m , c n,m , d n,m from the coefficients of the operators H 1 and H 2 is given in [8].…”

Three-fold symmetric Hahn-classical multiple orthogonal polynomials

“…However, the choice of the three-starlike sets is convenient because of the positivity of the measures and the location of the zeros. It turns out that the orthogonality measures under analysis are solutions to a second order differential equation and, as such, the recurrence coefficients for the whole set of the corresponding multiple orthogonal polynomials of type II can only be obtained algorithmically via the nearest-neighbor algorithm explained in [37] [19]. Some of the d-orthogonal polynomials that we found appear in the theory of random matrices, in particular in the investigation of singular values of products of Ginibre matrices, which uses multiple orthogonal polynomials with weight functions expressed in terms of Meijer G-functions [24].…”