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
The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. It is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information ...
uses the zeros of Stieltjes polynomials.
Definition
If ''P''
0, ''P''
1, form a sequence of
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the cl ...
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.
References
*{{eom, id=s/s120250, title=Stieltjes polynomials, first=Sven , last=Ehrich
Orthogonal polynomials