Quasi-orthogonality and zeros of some ₃ F ₂ hypergeometric polynomials

Abstract

The location of the zeros of general hypergeometric polynomials are linked with those of the classical orthogonal polynomials in some cases, notably ₂ F ₁ and ₁ F ₁ hypergeometric polynomials which have been extensively studied. In the case of ₃ F ₂ polynomials, less is known about their properties, including the location of their zeros, because there is, in general, no direct link with orthogonal polynomials. We consider two classes of ₃ F ₂ hypergeometric polynomials, each of which has a representation in terms of ₂ F ₁ polynomials. Our main result proves that the class of polynomials ₃ F ₂(– n, a, b;a – 1, d; x), a, b, d ∈ R, n ∈ N is quasi-orthogonal of order 1 on an interval that varies with the real parameters b and d. We discuss the apparent role played by the parameter a with regard to the location of all the zeros of this class of polynomials. We also prove results on the location of the zeros of the class ₃ F ₂(– n, b,(b – n)/2; b – n, (b – n – 1)/2; x), b ∈ R, n ∈N by using the orthogonality and quasi-orthogonality of factors involved in its representation.