Definitions

# Secondary polynomials

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

$q_n\left(x\right) = int_mathbb\left\{R\right\}! frac\left\{p_n\left(t\right) - p_n\left(x\right)\right\}\left\{t - x\right\} rho\left(t\right),dt.$

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

begin\left\{align\right\} q_0\left(x\right) &\left\{\right\}
= 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^2rho(t),dt + xint_mathbb{R} ! trho(t),dt + x^2int_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.