# .

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.