On a Conjecture Concerning Monotonicity of Zeros of Ultraspherical Polynomials Original Research Article

LetCλn,n=0, ;1, …, λ>−1/2 be the ultraspherical (Gegenbauer) polynomials, orthogonal on (−1, 1) with respect to the weight (1−x2)λ−1/2. Denote byζn, k(λ),k=1, …, [n/2] the positive zeros ofCλnenumerated in decreasing order. The problem of finding the “extremal” functionffor which the productsf(λ) ζn, k(λ) are increasing functions ofλis of recent interest. Ismail, Letessier, and Askey conjectured thatf(λ)=(λ+1)1/2is the function to solve this problem. We prove the conjecture for sufficiently largenand some related results.