Inequalities for zeros of Jacobi polynomials via Sturm's theorem: Gautschi's conjectures
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Springer
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Abstract
Let x(n,k)((alpha,beta)), k = 1, ... , n, be the zeros of Jacobi polynomials P-n((alpha,beta)) (x) arranged in decreasing order on (-1, 1), where alpha, beta > -1, and theta((alpha,beta))(n,k) = arccos x(n,k)((alpha,beta)). Gautschi, in a series of recent papers, conjectured that the inequalitiesn theta((alpha,beta))(n,k) < (n + 1)theta((alpha,beta))(n+1,k)and(n + (alpha + beta + 3)/2)theta((alpha,beta))(n+1,k) < (n + (alpha + beta + 1)/2)theta((alpha,beta))(n,k),hold for all n >= 1, k = 1, ... , n, and certain values of the parameters alpha and beta. We establish these conjectures for large domains of the (alpha, beta)-plane by using a Sturmian approach.
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Gautschi's conjectures, Jacobi polynomials, Zeros, Inequalities
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English
Citation
Numerical Algorithms. Dordrecht: Springer, v. 67, n. 3, p. 549-563, 2014.
