On connection between zeros and d -orthogonality

Abstract

AbstractConnection 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 {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, we give necessary and sufficient conditions on the zeros of a given sequence {Pn}n≥0, that will assure that this sequence satisfies a particular (d+1)-order recurrence relation. Many examples to illustrate the obtained results are given.KEYWORDS: (d+1)-Fold symmetric polynomialsinterlacing propertyd-orthogonal polynomialsrecurrence relationzeros of polynomialsAMS CLASSIFICATION:: 42C0533C45 AcknowledgmentsThe authors thank the anonymous referees for their helpful comments and suggestions that improved the quality of the manuscript.Disclosure statementNo potential conflict of interest was reported by the author(s).