Monotonicity Properties of the Zeros of Ultraspherical Polynomials Original Research Article

Letx(λ)n, k,k=1, 2, …, [n/2], denote thekth positive zero in increasing order of the ultraspherical polynomialP(λ)n(x). We prove that the function [λ+(2n2+1)/(4n+2)]1/2 x(λ)n, kincreases asλincreases forλ>−1/2. The proof is based on two integrals involved with the square of the ultraspherical polynomialP(λ)n(x).