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


 * $$ q_n(x) = \int_\mathbb{R}\! \frac{p_n(t) - p_n(x)}{t - x} \rho(t)\,dt. $$

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


 * $$\begin{align} q_0(x) &{}

= \int_\mathbb{R} \! \frac{t^3 - x^3}{t - x} \rho(t)\,dt \\ &{} = \int_\mathbb{R} \! \frac{(t - x)(t^2+tx+x^2)}{t - x} \rho(t)\,dt \\ &{} = \int_\mathbb{R} \! (t^2+tx+x^2)\rho(t)\,dt \\ &{} = \int_\mathbb{R} \! t^2\rho(t)\,dt + x\int_\mathbb{R} \! t\rho(t)\,dt + x^2\int_\mathbb{R} \! \rho(t)\,dt \end{align}$$

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