Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Abstract

Let ℝ=(−∞,∞), and let Q∈C2:ℝ→[0,∞) be an even function. In this paper, we consider the exponential-type weights wρ(x)=|x|ρexp(−Q(x)),  ρ>−1/2,  x∈ℝ, and the orthonormal polynomials pn(wρ2;x) of degree n with respect to wρ(x). So, we obtain a certain differential equation of higher order with respect to pn(wρ2;x) and we estimate the higher-order derivatives of pn(wρ2;x) and the coefficients of the higher-order Hermite-Fejér interpolation polynomial based at the zeros of pn(wρ2;x).