On the Behaviour of Zeros of Jacobi Polynomials Original Research Article

Denote by xn,k(α,β) and xn,k(λ)=xn,k(λ−1/2,λ−1/2) the zeros, in decreasing order, of the Jacobi polynomial P(α,β)n(x) and of the ultraspherical (Gegenbauer) polynomial Cλn(x), respectively. The monotonicity of xn,k(α,β) as functions of α and β, α,β>−1, is investigated. Necessary conditions such that the zeros of P(a,b)n(x) are smaller (greater) than the zeros of P(α,β)n(x) are provided. A. Markov proved that xn,k(a,b)xn,k(α,β)) for every n∈N and each k, 1⩽k⩽n if a>α and b<β (a<α and b>β). We prove the converse statement of Markovʹs theorem. The question of how large the function fn(λ) could be such that the products fn(λ)xn,k(λ), k=1,…,[n/2] are increasing functions of λ, for λ>−1/2, is also discussed. Elbert and Siafarikas proved that fn(λ)=(λ+(2n2+1)/(4n+2))1/2 obeys this property. We establish the sharpness of their result.