Zeros of quadratic quasi-orthogonal order 2 polynomials

Abstract

Corollary 2 in [1] states that for −32<λ<−12, n∈N, the quasi-orthogonal order 2 Gegenbauer polynomial Cn(λ)(x) has n−2 real, distinct zeros in (−1,1), one zero larger than 1 and one zero smaller than −1. This is correct provided n≥3, n∈N, but does not hold when n=2 for every λ in the range −32<λ<−12. An elementary calculation shows that the quasi-orthogonal order 2 Gegenbauer polynomial C2(λ)(x) has 2 real, distinct zeros with one zero larger than 1 and one zero smaller than −1 when −1<λ<−12 and two distinct pure imaginary zeros when −32<λ<−1. A similar error occurs in the proof of Corollary 4(i) in [1] relating to the location of the zeros of the quadratic quasi-orthogonal order 2 Jacobi polynomial P2(α,β)(x), −2<α,β<−1. Each error arises from a different incorrect application of Theorem VII due to Shohat (cf. [8, p. 472]). We discuss the Hilbert–Klein formulas (cf. [9, p. 145]) and indicate the overlap between two different stages of the migration process of the zeros of Cn(λ)(x) from the real axis to the imaginary axis (see [4], Section 3) that occurs when n=2.