Secondary polynomials

In mathematics, the secondary polynomials $\{q_n(x)\}$ associated with a sequence $\{p_n(x)\}$ of polynomials orthogonal with respect to a density $\rho (x)$ are defined by

$q_n(x)=\underset{\mathbb{ℝ}}{\int }\frac{p_n(t)-p_n(x)}{t-x}\rho (t)dt.$

To see that the functions $q_n(x)$ are indeed polynomials, consider the simple example of $p_0(x)=x^3.$ Then,

$\begin{array}{rl}{q}_0(x)& =\underset{\mathbb{ℝ}}{\int }\frac{t^3-x^3}{t-x}\rho (t)dt\\ & =\underset{\mathbb{ℝ}}{\int }\frac{(t-x)(t^2+tx+x^2)}{t-x}\rho (t)dt\\ & =\underset{\mathbb{ℝ}}{\int }(t^2+tx+x^2)\rho (t)dt\\ & =\underset{\mathbb{ℝ}}{\int }{t}^2\rho (t)dt+x\underset{\mathbb{ℝ}}{\int }t\rho (t)dt+x^2\underset{\mathbb{ℝ}}{\int }\rho (t)dt\end{array}$

which is a polynomial $x$ provided that the three integrals in $t$ (the moments of the density $\rho$) are convergent.

See also

• Secondary measure