On the zeros of Gonchar polynomials

Martin Lamprecht
2013 Proceedings of the American Mathematical Society  
We verify two conjectures of Brauchart et al. concerning the zeros of the Gonchar polynomials G(d; z) : In [1] the polynomials were baptized Gonchar polynomials, since they arise from an electrostatics problem on the sphere that was originally posed by A. A. Gonchar. Several interesting algebraic and asymptotic properties of the sequence G(d; z) were proven in [1], and in this note we will verify two conjectures from [1] concerning the location of the zeros of the Gonchar polynomials. Our first
more » ... result deals with the monotonicity of the largest positive zero ξ(d) of G(d; z) as a function of d ∈ N. As shown in [1, Prop. 7], ξ(d) lies in (2, 3] and is the only zero of G(d; z) there (multiplicities counted). The following was stated as Proposition 8 in [1]. Proposition 1. The sequence {ξ(d)} d∈N is strictly decreasing. Proof. It is easy to see that P (d; ξ(d)) = 0 is equivalent to
doi:10.1090/s0002-9939-2013-11866-6 fatcat:spggcprzh5f3bgue2hap7cprei