It is well known that the zeros of orthogonal polynomials interlace. In this
paper we study the case of multiple orthogonal polynomials. We recall known
results and some recursion relations for multiple orthogonal polynomials. Our
main result gives a sufficient condition, based on the coefficients in the
recurrence relations, for the interlacing of the zeros of neighboring multiple
orthogonal polynomials. We give several examples illustrating our result.Comment: 18 page
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.
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