Interlacing of the Zeros and d-Orthogonality

Abstract Connection between the interlacing of the zeros and the orthogonality of a given sequence of polynomials was 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 {Pn}n≥0, with the existence of a certain ratio cn expressed by means of the zeros of Pn. Then, for the (d + 1)-fold symmetric polynomials, {Pn}n≥0, such that Pn has qn distinct positive real zeros, n = (d + 1)qn + j, j = 0,...,d, we study the connection between the interlacing of these zeros, the d-orthogonality and the positivity of the ratio cn. Finally, for two other particular cases of d-orthogonal polynomials, we give necessary and sufficient conditions on the zeros that will assure that the sequence {Pn}n≥0, satisfies a particular (d + 1)-order recurrence relation. Many examples to illustrate the obtained results are given. Mathematics Subject Classification (2010). 42C05, 33C45.