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,

= 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.

See also

• Secondary measure