Inequalities for zeros of Jacobi polynomials via Sturm’s theorem: Gautschi’s conjectures

Let x n , k ( α , β ) , k = 1 , … , n , be the zeros of Jacobi polynomials P n ( α , β ) ( x ) arranged in decreasing order on ( − 1 , 1 ) , where α , β > − 1 , and θ n , k ( α , β ) = arccos x n , k ( α , β ) . Gautschi, in a series of recent papers, conjectured that the inequalities n θ n , k ( α , β ) < ( n + 1 ) θ n + 1 , k ( α , β ) and ( n + ( α + β + 3 ) / 2 ) θ n + 1 , k ( α , β ) < ( n + ( α + β + 1 ) / 2 ) θ n , k ( α , β ) , hold for all n ≥ 1 , k = 1 , … , n , and certain values of the parameters α and β . We establish these conjectures for large domains of the ( α , β ) -plane by using a Sturmian approach.