Secondary polynomials

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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{align} 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{align}

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

See also[edit]