Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and image-minimal polynomials, image Original Research Article

Let image be a sequence of polynomials with real coefficients such that image uniformly for image with image on image, where image and image. First it is shown that the zeros of image are dense in image, have spacing of precise order image and are interlacing with the zeros of image on image for every image. Let image be another sequence of real polynomials with image uniformly on image and image on image It is demonstrated that for all sufficiently large n the zeros of image and image strictly interlace on image if image on image. If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on image, image, is obtained. Finally it is shown that the results hold for wide classes of weighted image-minimal polynomials, image linear combinations and products of orthogonal polynomials, etc.