Estimates for the first and second derivatives of the Stieltjes polynomials

Abstract

Let w(lambda)(x) := (1 - x(2))(lambda-1/2) and P-n((lambda)) be the ultraspherical polynomials with respect to w(lambda)(x). Then we denote E-n+1((lambda)) the Stieltjes polynomials with respect to w(lambda)(x) satisfying [GRAPHICS] In this paper, we give estimates for the first and second derivatives of the Stieltjes polynomials E-n+1((lambda)) and the product E-n+1((lambda)) P-n((lambda)) by obtaining the asymptotic differential relations. Moreover, using these differential relations we estimate the second derivatives of E-n+1((lambda)) (x) and E-n+1((lambda)) (x) P-n((lambda)) (x) at the zeros of E-n+1((lambda)) (x) and the product E-n+1((lambda)) (x)P-n((lambda)) (x), respectively. (C) 2004 Elsevier Inc. All rights reserved.