On the Zeros of Various Kinds of Orthogonal Polynomials

Recently, several generalizations of the notion of orthogonal polynomials appeared in the literature. The aim of this paper is to study their zeros. Let P k be the unique polynomial of exact degree k such that Z b a x i P k (x) dd(x) = 0; for i = 0; : : : ; k ? 1 where is a positive Borel measure on a; b]. P k is the polynomial of degree k belonging to the family of orthogonal polynomials on a; b] with respect to the measure. It is well{known that the zeros of P k are all real, distinct and in a; b] and that they interlace with the zeros of P k+1. Recently, several generalizations of the notion of orthogonality appeared in the literature. This paper is devoted to the properties of their zeros. The deenitions and the rst results on the number of their real zeros will be given in Section 1. Since these rst results do not give a complete information on the zeros, some additional results will be proved in Section 2. The last section will be devoted to some vector generalizations of Chebyshev polynomials. The results presented in this paper are far to be complete. They are only given here as a starting point for further investigations. 1. Deenitions and rst results In this section, we shall deene the various generalizations of orthogonal polynomials that will be studied and we shall give a rst result on the number of their real zeros. All the results are based, in fact, on the notion of quasi{orthogonal polynomials that will be treated in Subsection 1.1. Then, the cases of biorthogonal, simultaneous orthogonal and vector orthogonal polynomials will be discussed. The questions of existence and uniqueness of these polynomials will not be considered here and the polynomials will always be assumed to be uniquely determined by the orthogonality conditions apart from a nonzero multiplying factor. The case of quasi{orthogonality will also be discussed. 1.1 Quasi{orthogonal polynomials Let L be a linear functional on the space of polynomials and let P k be a polynomial of exact degree