Some notes on the zeros of power orthogonal polynomials
One section of the paper is devoted to estimation of the largest zeros of power orthogonal polynomials. Here \(n\)th power orthogonal polynomial with respect to \(d\mu\) is a polynomial \(\omega_n(d\mu,{\mathbf m};x)=(x-x_{1n})\ldots(x-x_{nn})\) such that \[ \Phi_n(x_{1n},\ldots,x_{nn})=\inf_{{\mathbf y}\in\bar Y}\Phi_n(y_1,\ldots,y_n), \] where \(\Phi_n(y_1,\ldots,y_n)=\int_I\Omega_n({\mathbf y};x)d\mu(x),\) \[ \Omega_n({\mathbf y};x)=\prod_{k=1}^n|x-y_k|^{m_k},\quad Y=\{{\mathbf y}=(y_1,\ldots,y_n):a<y_1<y_2<\ldots<y_n<b\}, \] \({\mathbf m}=(m_1,\ldots,m_n), I=(a,b),-\infty\leq a<b\leq\infty.\) Another section contains results about usual orthogonal polynomials including inequalities involving the largext zeros and the Turán inequality.
scientific article; zbMATH DE number 5376867
scientific article