Christoffel Transform and Multiple Orthogonal Polynomials

Abstract: We identify a connection between the Christoffel transform of orthogonal polynomials and multiple orthogonality systems containing a finitely supported measure. In consequence, the compatibility relations for the nearest neighbour recurrence coefficients provide a new algorithm for the computation of the Jacobi coefficients of the one-step or multi-step Christoffel transforms. More generally, we investigate multiple orthogonal polynomials associated with the system of measures obtained by applying a Christoffel transform to each of the orthogonality measures. We present an algorithm for computing the transformed recurrence coefficients, and determinantal formulas for the transformed multiple orthogonal polynomials of type I and type II. Finally, we show that zeros of multiple orthogonal polynomials of an Angelesco or an AT system interlace with the zeros of the polynomial corresponding to the one-step Christoffel transform. This allows us to prove a number of interlacing properties satisfied by the multiple orthogonality analogues of classical orthogonal polynomials. For the discrete polynomials, this also produces an estimate on the smallest distance between consecutive zeros.