Secondary polynomials

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In mathematics, the secondary polynomials ${q n (x)}$ associated with a sequence ${p n (x)}$ of polynomials orthogonal with respect to a density $ρ(x)$ 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,

$\begin{align} q_0(x) &{} = \int_\mathbb{R} \! \frac{t^3 - x^3}{t - x} \rho(t)\,dt \\ &{} = \int_\mathbb{R} \! \frac{(t - x)(t^2+tx+x^2)}{t - x} \rho(t)\,dt \\ &{} = \int_\mathbb{R} \! (t^2+tx+x^2)\rho(t)\,dt \\ &{} = \int_\mathbb{R} \! t^2\rho(t)\,dt + x\int_\mathbb{R} \! t\rho(t)\,dt + x^2\int_\mathbb{R} \! \rho(t)\,dt \end{align}$