Zeros of polynomials orthogonal over regular \(N\)-gons.
The authors investigate the location of zeros of Bergman polynomials \(Q_n(z;G_N)\) (orthogonal polynomials with respect to area measure) for regular \(N\)-gons \(G_N\) in the plane. Two conjectures posed by Eiermann and Stahl are proved: (I) For \(N = 3, 4\), the zeros of all the \(Q_n\)'s are located exactly on the `diagonals' \(\Gamma_{k,N}\) of \(G_N\). However, for \(N\geqslant5\) there are zeros that are not on the \(\Gamma_{k,N}\)'s. (II) For \(N = 3, 4\) and fixed \(j\in\{0,\dots, N- 1\}\), the real zeros of the \(Q_{Nl+j}\)'s, \(l=0,1,\dots\), interlace on \((0,1)\). Some consequences regarding the asymptotic behavior of Bergman polynomials are given.
location of zeros of Bergman polynomials
asymptotic behavior of Bergman polynomials
scientific article; zbMATH DE number 1924736