On the Complex Zeros of Some Families of Orthogonal Polynomials
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 polynomia...
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2010-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/263860 |
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doaj-625ac9bff2ed4bd992c8c95b0822abaf2020-11-24T22:26:03ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092010-01-01201010.1155/2010/263860263860On the Complex Zeros of Some Families of Orthogonal PolynomialsEugenia N. Petropoulou0Division of Applied Mathematics and Mechanics, Department of Engineering Sciences, University of Patras, 26500 Patras, GreeceThe 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.http://dx.doi.org/10.1155/2010/263860 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Eugenia N. Petropoulou |
| spellingShingle |
Eugenia N. Petropoulou On the Complex Zeros of Some Families of Orthogonal Polynomials Abstract and Applied Analysis |
| author_facet |
Eugenia N. Petropoulou |
| author_sort |
Eugenia N. Petropoulou |
| title |
On the Complex Zeros of Some Families of Orthogonal Polynomials |
| title_short |
On the Complex Zeros of Some Families of Orthogonal Polynomials |
| title_full |
On the Complex Zeros of Some Families of Orthogonal Polynomials |
| title_fullStr |
On the Complex Zeros of Some Families of Orthogonal Polynomials |
| title_full_unstemmed |
On the Complex Zeros of Some Families of Orthogonal Polynomials |
| title_sort |
on the complex zeros of some families of orthogonal polynomials |
| publisher |
Hindawi Limited |
| series |
Abstract and Applied Analysis |
| issn |
1085-3375 1687-0409 |
| publishDate |
2010-01-01 |
| description |
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. |
| url |
http://dx.doi.org/10.1155/2010/263860 |
| work_keys_str_mv |
AT eugenianpetropoulou onthecomplexzerosofsomefamiliesoforthogonalpolynomials |
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