On a Conjecture Concerning Monotonicity of Zeros of Ultraspherical Polynomials
نویسندگان
چکیده
منابع مشابه
Monotonicity of zeros of Jacobi polynomials
Denote by xn,k(α, β), k = 1, . . . , n, the zeros of the Jacobi polynomial P (α,β) n (x). It is well known that xn,k(α, β) are increasing functions of β and decreasing functions of α. In this paper we investigate the question of how fast the functions 1 − xn,k(α, β) decrease as β increases. We prove that the products tn,k(α, β) := fn(α, β) (1− xn,k(α, β)), where fn(α, β) = 2n2 + 2n(α + β + 1) +...
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We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial C n 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 a...
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1. It will be recalled that the ultraspherical polynomials are those which are orthogonal on the interval ( — 1, 1), corresponding to the weight function (1— x2)x~1/2, X>—1/2. In what follows X = 0 will also be excluded. The coefficients of these polynomials are functions of the parameter X appearing in the weight function, and the symbol P„(x, X), indicative of this fact, will be used to denot...
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ژورنال
عنوان ژورنال: Journal of Approximation Theory
سال: 1996
ISSN: 0021-9045
DOI: 10.1006/jath.1996.0030