Freud equations for Legendre polynomials on a circular arc and solution of the Grünbaum–Delsarte–Janssen–Vries problem

One establishes inequalities for the coefficients of orthogonal polynomialsΦn(z)=zn+ξnzn-1+⋯+Φn(0),n=0,1,…,which are orthogonal with respect to a constant weight on the arc of the unit circle S={eiθ,απ<θ<2π-απ}S={eiθ,απ<θ<2π-απ}, with 0<α<10<α<1. Recurrence relations (Freud equations), and differential relations are used. Among other results, it is shown that Φn(0)>0,n=1,2,…Φn(0)>0,n=1,2,…