## Abstract and Applied Analysis

### On the Complex Zeros of Some Families of Orthogonal Polynomials

Eugenia N. Petropoulou

#### Abstract

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

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 263860, 14 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620737

Digital Object Identifier
doi:10.1155/2010/263860

Mathematical Reviews number (MathSciNet)
MR2660396

Zentralblatt MATH identifier
1200.30002

#### Citation

Petropoulou, Eugenia N. On the Complex Zeros of Some Families of Orthogonal Polynomials. Abstr. Appl. Anal. 2010 (2010), Article ID 263860, 14 pages. doi:10.1155/2010/263860. https://projecteuclid.org/euclid.aaa/1288620737