Stieltjes polynomials

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 "P"0, "P"1, 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.