On the complex zeros of some families of orthogonal polynomials (Q1957565)

Summary: The complex zeros of the orthogonal Laguerre polynomials \(L_n^{(a)}(x)\) for \(a<-n\), ultraspherical polynomials \(P_n^{(\lambda)}(x)\) for \(\lambda<-n\), Jacobi polynomials \(P_n^{(a,\beta)}(x)\) for \(a<-n\), \(\beta<-n\), \(a+\beta<-2(n+1)\), orthonormal Al-Salam-Carlitz II polynomials \(P_n^{(a)}(x;q)\) for \(a<0\), \(0<q<1\), and \(q\)-Laguerre polynomials \(L_n^{(a)}(x;q)\) for \(a<-n\), \(0<q<1\) are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.