On the Complex Zeros of Some Families of Orthogonal Polynomials

Abstract

The complex zeros of the orthogonal Laguerre polynomials $L_n^{\left(a\right)}\left(x\right)$ for $a<-n$, ultraspherical polynomials $P_n^{\left(\lambda \right)}\left(x\right)$ for $\lambda <-n$, Jacobi polynomials $P_n^{(a,\beta )}\left(x\right)$ for $a<-n$, $\beta <-n$, $a+\beta <-2(n+1)$, orthonormal Al-Salam-Carlitz II polynomials $P_n^{\left(a\right)}(x;q)$ for $a<0$, $0<q<1$, and $q$-Laguerre polynomials $L_n^{\left(a\right)}(x;q)$ 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.