Zeros of Quasi-Orthogonal Jacobi Polynomials
We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by α>−1, −2<β<−1. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials P(α,β)n and P(α,β+2)n are interla...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2016 |
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| Sprache: | English |
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Інститут математики НАН України
2016
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147743 |
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| Zitieren: | Zeros of Quasi-Orthogonal Jacobi Polynomials / K. Driver, K. Jordaan // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. |
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Driver, K. Jordaan, K. 2019-02-15T19:02:14Z 2019-02-15T19:02:14Z 2016 Zeros of Quasi-Orthogonal Jacobi Polynomials / K. Driver, K. Jordaan // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C50; 42C05 DOI:10.3842/SIGMA.2016.042 https://nasplib.isofts.kiev.ua/handle/123456789/147743 We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by α>−1, −2<β<−1. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials P(α,β)n and P(α,β+2)n are interlacing, holds when the parameters α and β are in the range α>−1 and −2<β<−1. We prove that the zeros of P(α,β)n and P(α,β)n₊₁ do not interlace for any n∈N, n≥2 and any fixed α, β with α>−1, −2<β<−1. The interlacing of zeros of P(α,β)n and P(α,β+t)m for m,n∈N is discussed for α and β in this range, t≥1, and new upper and lower bounds are derived for the zero of P(α,β)n that is less than −1. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html. The research of both authors was funded by the National Research Foundation of South Africa. We thank the referees for helpful suggestions and insights. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Zeros of Quasi-Orthogonal Jacobi Polynomials Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Zeros of Quasi-Orthogonal Jacobi Polynomials |
| spellingShingle |
Zeros of Quasi-Orthogonal Jacobi Polynomials Driver, K. Jordaan, K. |
| title_short |
Zeros of Quasi-Orthogonal Jacobi Polynomials |
| title_full |
Zeros of Quasi-Orthogonal Jacobi Polynomials |
| title_fullStr |
Zeros of Quasi-Orthogonal Jacobi Polynomials |
| title_full_unstemmed |
Zeros of Quasi-Orthogonal Jacobi Polynomials |
| title_sort |
zeros of quasi-orthogonal jacobi polynomials |
| author |
Driver, K. Jordaan, K. |
| author_facet |
Driver, K. Jordaan, K. |
| publishDate |
2016 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by α>−1, −2<β<−1. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials P(α,β)n and P(α,β+2)n are interlacing, holds when the parameters α and β are in the range α>−1 and −2<β<−1. We prove that the zeros of P(α,β)n and P(α,β)n₊₁ do not interlace for any n∈N, n≥2 and any fixed α, β with α>−1, −2<β<−1. The interlacing of zeros of P(α,β)n and P(α,β+t)m for m,n∈N is discussed for α and β in this range, t≥1, and new upper and lower bounds are derived for the zero of P(α,β)n that is less than −1.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147743 |
| citation_txt |
Zeros of Quasi-Orthogonal Jacobi Polynomials / K. Driver, K. Jordaan // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 27 назв. — англ. |
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2025-12-07T17:40:27Z |
| last_indexed |
2025-12-07T17:40:27Z |
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1850872144437706752 |