On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

Abstract

Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality (equation presented) holds for α, β > - 1 and n ≥ 1, θ∈ ∈(0, π), where Pn(α,β)(x) are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where (α, β) ∈ {(1/2,1/2), (1/2,-1/2), (-1/2,1/2}}. © 2007 Springer Science+Business Media LLC.