On the Complex Zeros of Some Families of Orthogonal Polynomials

Abstract

The complex zeros of the orthogonal Laguerre polynomials 𝐿𝑛(𝑎)(𝑥) for 𝑎<−𝑛, ultraspherical polynomials 𝑃𝑛(𝜆)(𝑥) for 𝜆<−𝑛, Jacobi polynomials 𝑃𝑛(𝑎,𝛽)(𝑥) for 𝑎<−𝑛, 𝛽<−𝑛, 𝑎+𝛽<−2(𝑛+1), orthonormal Al-Salam-Carlitz II polynomials 𝑃𝑛(𝑎)(𝑥;𝑞) for 𝑎<0, 0<𝑞<1, and 𝑞-Laguerre polynomials 𝐿𝑛(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.