On connection between zeros and d-orthogonality

Abstract

Connection between the interlacing of the zeros and the orthogonality of a given sequence of polynomials is done by K. Driver. In this paper, we attempt to extend this result to some particular cases of d-orthogonal polynomials. In fact, first, we characterize the 2-orthogonality of a given sequence , with the existence of a certain ratio expressed by means of the zeros of . Then, for the -fold symmetric polynomials, , such that has distinct positive real zeros, , we study the connection between the interlacing of these zeros, the d-orthogonality and the positivity of the ratio . Finally, we give necessary and sufficient conditions on the zeros of a given sequence , that will assure that this sequence satisfies a particular -order recurrence relation. Many examples to illustrate the obtained results are given.