Stieltjes polynomials

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For the orthogonal polynomials, see Stieltjes–Wigert polynomials.
For the Stieltjes polynomial solutions of Fuchsian differential equations, see Heine–Stieltjes polynomials.

In mathematics, the Stieltjes polynomials En 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[edit]

If P0, P1, form a sequence of orthogonal polynomials for some inner product, then the Stieltjes polynomial En is a degree n polynomial orthogonal to Pn–1(x)xk for k = 0, 1, ..., n – 1.

References[edit]