On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

摘要

Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality $frac{P_{n}^{(alpha,beta)}(cosfrac{theta}{n})}{P_{n}^{(alpha,beta)}(1)} ? 1 and n ≥ 1, θ ∈ (0, π), where $P_{n}^{(alpha,beta)}(x)$ are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where $(alpha, ,beta)in big{(frac{1}{2},frac{1}{2}),,(frac{1}{2},-frac{1}{2}),,(-frac{1}{2},frac{1}{2})big}$ .