Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

• Authors: Dimitar K. Dimitrov
• Venue: Journal of Computational and Applied Mathematics - Proceedings of the sixth international symposium on orthogonal polynomials, special functions and their applications
• Year: 2003

Let Cnλ(x), n=0, 1,....,λ -½, be the ultraspherical (Gegenbauer) polynomials, orthogonal in (-1, 1) with respect to the weight function (1 - x2)λ-1/2. Denote by xnk(λ), k = 1,.... ,n, the zeros of Cnλ(x) enumerated in decreasing order. In this short note, we prove that, for any n ∈ N, the product (λ + 1)3/2xn1(λ) is a convex function of λ if λ ≥ 0. The result is applied to obtain some inequalities for the largest zeros of Cnλ(x). If xnk(α), k = 1,..... n, are the zeros of Laguerre polynomial Lnα(x), also enumerated in decreasing order, we prove that xn1(λ)/(α + 1) is a convex function of α for α - 1.