Corollary 2 in [1] states that for $-rac{3}{2} < lambda < -rac{1}{2}, n in mathbb{N}$, the quasi-orthogonal order 2 Gegenbauer polynomial $C_n^{(lambda)}(x)$ has $n-2$ real, distinct zeros in $(-1,1),$ one zero larger than $1$ and one zero smaller than $-1.$ This is correct provided $n geq 3,$ $n in mathbb{N},$ but does not hold when $n=2$ for every $lambda$ in the range $-rac{3}{2} < lambda < -rac{1}{2}.$ An elementary calculation shows that the quasi-orthogonal order $2$ Gegenbauer polynomial $C_2^{(lambda)}(x)$ has $2$ real, distinct zeros with one zero larger than $1$ and one zero smaller than $-1$ when $-1 < lambda < -rac{1}{2}$ and two distinct pure imaginary zeros when $-rac{3}{2} < lambda < -1.$ 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 $P_{2}^{(alpha,eta)}(x)$, $-2 < alpha, eta < -1.$ 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 $C_n^{(lambda)}(x)$ from the real axis to the imaginary axis (see [4] Section 3) that occurs when $n=2.$

Zeros of quadratic quasi-orthogonal order 2 polynomials

Redivo-Zaglia, Michela
2019

Abstract

Corollary 2 in [1] states that for $-rac{3}{2} < lambda < -rac{1}{2}, n in mathbb{N}$, the quasi-orthogonal order 2 Gegenbauer polynomial $C_n^{(lambda)}(x)$ has $n-2$ real, distinct zeros in $(-1,1),$ one zero larger than $1$ and one zero smaller than $-1.$ This is correct provided $n geq 3,$ $n in mathbb{N},$ but does not hold when $n=2$ for every $lambda$ in the range $-rac{3}{2} < lambda < -rac{1}{2}.$ An elementary calculation shows that the quasi-orthogonal order $2$ Gegenbauer polynomial $C_2^{(lambda)}(x)$ has $2$ real, distinct zeros with one zero larger than $1$ and one zero smaller than $-1$ when $-1 < lambda < -rac{1}{2}$ and two distinct pure imaginary zeros when $-rac{3}{2} < lambda < -1.$ 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 $P_{2}^{(alpha,eta)}(x)$, $-2 < alpha, eta < -1.$ 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 $C_n^{(lambda)}(x)$ from the real axis to the imaginary axis (see [4] Section 3) that occurs when $n=2.$
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3279652
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