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 $\frac{P_{n}^{(\alpha,\beta)}(\cos\frac{\theta}{n})}{P_{n}^{(\alpha,\beta)}(1)}<\frac{P_{n+1}^{(\alpha,\beta)}(\cos\frac{\theta}{n+1})}{P_{n+1}^{(\alpha,\beta)}(1)}$ holds for α,β > − 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 $(\alpha, \,\beta)\in \big\{(\frac{1}{2},\frac{1}{2}),\,(\frac{1}{2},-\frac{1}{2}),\,(-\frac{1}{2},\frac{1}{2})\big\}$.