Stieltjes polynomials explained

In mathematics, the Stieltjes polynomials E_n are polynomials associated to a family of orthogonal polynomials P_n. They are unrelated to the Stieltjes polynomial solutions of differential equations. Stieltjes originally considered the case where the orthogonal polynomials P_n are the Legendre polynomials.

The Gauss–Kronrod quadrature formula uses the zeros of Stieltjes polynomials.

Definition

If P₀, P₁, form a sequence of orthogonal polynomials for some inner product, then the Stieltjes polynomial E_n is a degree n polynomial orthogonal to P_n–1(x)x^k for k = 0, 1, ..., n – 1.