On the Complex Zeros of Some Families of Orthogonal Polynomials

and Applied Analysis 3 real and there is a need to locate their position. Moreover, the usual methods M1 – M3 mentioned before for the study of the zeros of Pn x may not apply at all, when Pn x are complex, or they may need serious modifications. Instead, the M4 method can be used directly. Such a functional analytic methodwas introduced in 10 andwas successfully used in a series of papers by the authors of 10 and their collaborators, including paper 11 , where results were given regarding the real part of the complex zeros of a class of polynomials including the generalized Bessel polynomials. The most recent application of this method was in 12, 13 , where convexity results and differential inequalities were deduced for the largest and lowest zeros and functions involving these zeros of several q-polynomials. This method is also used in the present paper and it is briefly presented in Section 3. Themain idea is to transform the problem of the zeros of Pn x satisfying 1.2 to the equivalent problem of the eigenvalues of a specific tridiagonal operator T . Then, by utilizing the properties of T , several properties of the zeros of Pn x can be proved. The aim of the present paper is to provide regions in C of the location of the complex zeros of the following: i Laguerre orthogonal polynomials L a n x for a < −n, ii ultraspherical orthogonal polynomials P λ n x for λ < −n, iii Jacobi orthogonal polynomials P a,β n x for a < −n, β < −n, a β < −2 n 1 , iv orthonormal Al-Salam-Carlitz II polynomials P a n x; q for a < 0, 0 < q < 1, v q-Laguerre orthonormal polynomials L a n x; q for a < −n, 0 < q < 1. These regions are given in the form of inequalities regarding the real and imaginary properties of the zeros of the polynomials under consideration. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are given. All these results are stated in Section 2 and proved in Section 4. The reason for choosing the above mentioned five classes of orthogonal polynomials, apart from pure mathematical curiosity, is the fact that their zeros and especially the zeros of the Jacobi and Laguerre polynomials admit a very interesting electrostatic interpretation see, e.g., 14–16 , 6, page 140 and the references therein . To the best of the author’s knowledge there are very few results concerning the location of the complex zeros of the classical or q-polynomials or their limit relations. More precisely, in the thesis 17 and the paper 18 , the behavior of the complex zeros of the Laguerre, q-Laguerre, and Jacobi polynomials is primarily studied. Among others, an inequality regarding the real part of the zeros of the Laguerre polynomials and limit relations regarding the zeros of the Laguerre, q-Laguerre and Jacobi polynomials are proved using their explicit formulae and their recurrence relations. Also in 19 , the zeros of the hypergeometric polynomial F −n, b; 2b; z , for b > −1/2 are studied. These results are then applied in order to obtain information for the zeros of the Ultraspherical for λ < −n and Jacobi for β − 1/2 a 1 − n, a > −2 and for a −2β − 2n − 1, β > −1 polynomials. Finally in 20 , the zeros of the Ultraspherical polynomials are further investigated. More precisely, the authors give a description of the trajectories of the zeros as λ decreases from −1/2 to 1 − n. Several useful figures created using Mathematica illustrate these trajectories when n 8. In the end, the authors conclude that “as λ descends below −7, all 8 zeros of P 8 x are on the imaginary axis tending symmetrically to the origin as λ → −∞”. 4 Abstract and Applied Analysis The results of the present paper specifically Theorems 2.1 and 2.4 complement and improve the results of 17, 19, 20 . 2. Main Results In this section, several theorems are stated regarding the complex zeros of the orthogonal Laguerre, Ultraspherical and Jacobi, as well as the orthonormal Al-Salam-Carlitz II and qLaguerre polynomials. In each case, a region of the complex plane is given where these zeros lie, as well as a few limit relations regarding their asymptotic behavior. The proofs of these theorems are given in Section 4. Theorem 2.1. The zeros xnk a Re xnk a i Im xnk a of the Laguerre orthogonal polynomials L a n x for a < −n satisfy the following relations: a 1 ≤ Re xnk a ≤ 2n a − 1, 2.1 |Im xnk a | ≤ 2 √ −n a 1 . 2.2 Moreover, lim a→−∞ xnk a a 1. 2.3 Remark 2.2. It is obvious from 2.1 that if a < 1 − 2n, then Re xnk a ≤ 0. Remark 2.3. In 17, pages 112–131 , using the explicit formula for the Laguerre polynomials and their recurrence relation, the inequality 2.1 was obtained, among other interesting relations. Moreover it was proved that lim a→−∞ ∣