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Stieltjes Polynomial

Stieltjes Polynomial

Definition

Let $P_{n}$ be Legendre polynomial defined by

\[\begin{eqnarray} \int_{-1}^{-1} P_{n}(x)x^{k} \ dx & = & 0 \quad k = 0, 1, \ldots, n - 1, \nonumber \\ P_{n}(1) & = & 1 \nonumber . \end{eqnarray}\]

For $n \ge 0$, Stieltjes polynomial $E_{n+1}$ is defined by

\[\begin{eqnarray} \int_{-1}^{1} P_{n}(x) E_{n+1}(x) x^{k} \ dx & = & 0 \label{stieltjes_polynomial_constraint_01} \\ E_{n+1}(x) & = & \frac{ 2^{n} }{ \gamma_{n} } x^{n+1} + p(x) \label{stieltjes_polynomial_constraint_02} \\ \gamma_{n} & = & \frac{ 2^{2n} (n!)^{2} }{ (2n + 1)! } \nonumber \\ \Pi_{n} & := & \left\{ \sum_{i=0}^{n} a_{i}x^{i} \mid a_{i} \in \mathbb{R} \right\} \nonumber \\ p & \in & \Pi_{n} \nonumber \end{eqnarray}\]

Stieltjes polynomial is othogonal to $P_{n}$ by weigthing $x^{k}$. Up to a multiplicative constant, the polynomial $E_{n+1}$ is defined uniquely by \(\eqref{stieltjes_polynomial_constraint_01}\).

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