Estimates for the first and second derivatives of the Stieltjes polynomials

Let w_λ(x)(1−x²)^λ−1/2 and P_n^(λ) be the ultraspherical polynomials with respect to w_λ(x). Then we denote E_n+1^(λ) the Stieltjes polynomials with respect to w_λ(x) satisfyingIn this paper, we give estimates for the first and second derivatives of the Stieltjes polynomials E_n+1^(λ) and the product E_n+1^(λ)P_n^(λ) by obtaining the asymptotic differential relations. Moreover, using these differential relations we estimate the second derivatives of E_n+1^(λ)(x) and E_n+1^(λ)(x)P_n^(λ)(x) at the zeros of E_n+1^(λ)(x) and the product E_n+1^(λ)(x)P_n^(λ)(x), respectively.