Estimates for the first and second derivatives of the Stieltjes polynomials

Let wλ(x) = (1 - x2)λ-1/2 and Pn(λ) be the ultraspherical polynomials with respect to wλ(x). Then we denote En-1(λ) the Stieltjes polynomials with respect to wλ(x) satisfying ∫11wλ(x) Pn(λ)(x)En+1(λ)(x)xmdx { =0, 0 ≤ m ≤ n + 1, ≠ = 0, m = n + 1. In this paper, we give estimates for the first and second derivatives of the Stielijes polynomials En-1(λ) and the product En+1(λ)Pn(λ) by obtaining the asymptotic differential relations. Moreover, using these differential relations we estimate the second derivatives of En+1(λ)(x) and En+1(λ)(x)Pn(λ)(x) at the zeros of En+1(λ)(x) and the product En+1(λ)(x)Pn(λ)(x) respectively.

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