From Wikipedia, the free encyclopedia
In
mathematics, the secondary polynomials
associated with a
sequence
of
polynomials
orthogonal with respect to a density
are defined by
![{\displaystyle q_{n}(x)=\int _{\mathbb {R} }\!{\frac {p_{n}(t)-p_{n}(x)}{t-x}}\rho (t)\,dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac769cb45fb6fe3e4b19b78c7170202c71fdc713)
To see that the functions
are indeed polynomials, consider the simple example of
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819e180e510fb0734b15cf7a80ea89e55f009138)
which is a polynomial
provided that the three integrals in
(the
moments of the density
) are convergent.
See also