Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials

Abstract

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 $-3/2<\lambda <\infty$.

10 pages, 1 figure