Estimates for the first and second derivatives of the Stieltjes polynomials

Let <f>wλ(x)≔(1-x2)λ-1/2</f> and <f>Pn(λ)</f> be the ultraspherical polynomials with respect to <f>wλ(x)</f>. Then we denote <f>En+1(λ)</f> the Stieltjes polynomials with respect to <f>wλ(x)</f> satisfyingIn this paper, we give estimates for the first and second derivatives of the Stieltjes polynomials <f>En+1(λ)</f> and the product <f>En+1(λ)Pn(λ)</f> by obtaining the asymptotic differential relations. Moreover, using these differential relations we estimate the second derivatives of <f>En+1(λ)(x)</f> and <f>En+1(λ)(x)Pn(λ)(x)</f> at the zeros of <f>En+1(λ)(x)</f> and the product <f>En+1(λ)(x)Pn(λ)(x)</f>, respectively. [Copyright &y& Elsevier]