Zeros of Quasi-Orthogonal Jacobi Polynomials

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2016
• Hauptverfasser: Driver, K., Jordaan, K.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут математики НАН України 2016
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/147743
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• 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.

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.