Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials

Autor:	Driver, Kathy, Muldoon, Martin E.
Rok vydání:	2016
Předmět:	Mathematics - Classical Analysis and ODEs 33C45
Druh dokumentu:	Working Paper
Popis:	We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial $C_{n}^{(\lambda)}$ that is greater than $1$ when $-3/2 < \lambda < -1/2.$ Our first approach uses mixed three term recurrence relations and interlacing of zeros while the second approach uses a method going back to Euler and Rayleigh and already applied to Bessel functions and Laguerre and $q$-Laguerre polynomials. We use the bounds obtained by the second method to simplify the proof of the interlacing of the zeros of $(1-x^2)C_{n}^{(\lambda)}$ and $C_{n+1}^{(\lambda)}$, for $-3/2 < \lambda < \infty$. Comment: 10 pages, 1 figure
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1602.00044 Zobrazit plný text záznamu View this record from Arxiv