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

Stieltjes polynomial

In mathematics, the Heine–Stieltjes polynomials or Stieltjes polynomials, introduced by , are polynomial solutions of a second-order Fuchsian equation, a differential equation all of whose singularities are regular...

 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

Orthogonal polynomials

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...

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.