On connection between zeros and d-orthogonality

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 {P-n}(n=0), with the existence of a certain ratio c(n) expressed by means of the zeros of P-n. Then, for the (d + 1)-fold symmetric polynomials, {P-n}(n=0), such that Pn has q(n) distinct positive real zeros, n = (d + 1)q(n) +j,j = 0, ... , d, we study the connection between the interlacing of these zeros, the d-orthogonality and the positivity of the ratio c(n). Finally, we give nec-essary and sufficient conditions on the zeros of a given sequence {{P-n}(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.

Automated summary

This paper discusses the connection between the interlacing of zeros and the orthogonality of a given sequence of polynomials, focusing on particular cases of d-orthogonal polynomials. The authors characterize the 2-orthogonality of the sequence by the existence of a certain ratio expressed in terms of the zeros. They also study the interlacing of zeros, d-orthogonality, and positivity of the ratio for (d + 1)-fold symmetric polynomials. Necessary and sufficient conditions for a given sequence to satisfy a particular (d + 1)-order recurrence relation are provided, along with illustrative examples.