In mathematics, the Stieltjes polynomials E<sub>n</sub> are polynomials associated to a family of orthogonal polynomials P<sub>n</sub>. They are unrelated to the Stieltjes polynomial solutions of differential equations. Stieltjes originally considered the case where the orthogonal polynomials P<sub>n</sub> are the Legendre polynomials.
The GaussâÂÂKronrod quadrature formula uses the zeros of Stieltjes polynomials.
If P<sub>0</sub>, P<sub>1</sub>, form a sequence of orthogonal polynomials for some inner product, then the Stieltjes polynomial E<sub>n</sub> is a degree n polynomial orthogonal to P<sub>nâÂÂ1</sub>(x)x<sup>k</sup> for k = 0, 1, ..., n â 1.