Secondary polynomials

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In mathematics, the secondary polynomials ${\displaystyle \{q_{n}(x)\}}$ associated with a sequence ${\displaystyle \{p_{n}(x)\}}$ of polynomials orthogonal with respect to a density ${\displaystyle \rho (x)}$ are defined by

${\displaystyle q_{n}(x)=\int _{\mathbb {R} }\!{\frac {p_{n}(t)-p_{n}(x)}{t-x}}\rho (t)\,dt.}$^[1]

To see that the functions ${\displaystyle q_{n}(x)}$ are indeed polynomials, consider the simple example of ${\displaystyle p_{0}(x)=x^{3}.}$ Then,

${\displaystyle {\begin{aligned}q_{0}(x)&{}=\int _{\mathbb {R} }\!{\frac {t^{3}-x^{3}}{t-x}}\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!{\frac {(t-x)(t^{2}+tx+x^{2})}{t-x}}\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!(t^{2}+tx+x^{2})\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!t^{2}\rho (t)\,dt+x\int _{\mathbb {R} }\!t\rho (t)\,dt+x^{2}\int _{\mathbb {R} }\!\rho (t)\,dt\end{aligned}}}$

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

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