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.

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

The Gauss–Kronrod quadrature formula uses the zeros of Stieltjes polynomials.

Definition

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
  • Ehrich, Sven (2001) [1994], "Stieltjes polynomials", Encyclopedia of Mathematics, EMS Press


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