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# Stieltjes polynomials

In mathematics, the Stieltjes polynomials *E*_{n} are polynomials associated to a family of orthogonal polynomials Pn. They are unrelated to the Stieltjes polynomial solutions of differential equations. Stieltjes originally considered the case where the orthogonal polynomials Pn 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.

References

Ehrich, Sven (2001), "Stieltjes polynomials", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104

When k = n or n−1 we saw in the previous section that Vk(Fn) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table at the right.

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