Monotonicity of zeros of Jacobi polynomials

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Denote by xnk (α, β), k = 1, ..., n, the zeros of the Jacobi polynomial Pn(α, β) (x). It is well known that xnk (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 - xnk (α, β) decrease as β increases. We prove that the products tnk (α, β) {colon equals} fn (α, β) fenced(1 - xnk (α, β)), where fn (α, β) = 2 n2 + 2 n (α + β + 1) + (α + 1) (β + 1) are already increasing functions of β and that, for any fixed α > - 1, fn (α, β) is the asymptotically extremal, with respect to n, function of β that forces the products tnk (α, β) to increase. © 2007 Elsevier Inc. All rights reserved.

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Dimitrov, D. K., & Rafaeli, F. R. (2007). Monotonicity of zeros of Jacobi polynomials. Journal of Approximation Theory, 149(1), 15–29.

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