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{aligned}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{aligned}}

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

See also