Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials

TL;DR

This paper derives bounds for the positive zeros greater than one of ultraspherical polynomials in a specific parameter range, comparing two methods and applying results to zero interlacing properties.

Contribution

It introduces and compares two methods for bounding zeros of ultraspherical polynomials, simplifying proofs of zero interlacing for certain parameter ranges.

Findings

Bounds for zeros are established using two different methods.

The second method's bounds simplify proofs of zero interlacing.

Interlacing of zeros is confirmed for a broad parameter range.

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