Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials

2016/01/30 by Driver, Kathy, Muldoon, Martin E.
#33C45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1602.00044

Abstract

We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial C_n^(λ) that is greater than 1 when -3/2 < λ< -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-x²)C_n^(λ) and C_n+1^(λ), for -3/2 < λ< ∞.

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