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arXiv · 1602.00044

Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials

Kathy Driver·Martin E. Muldoon
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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 $-3/2 < \lambda < \infty$.

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Kathy Driver, Martin E. Muldoon. 2016-01-30. Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials. https://arxiv.org/abs/1602.00044

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