Secondary polynomials

In mathematics, the secondary polynomials [math]\displaystyle{ \{q_n(x)\} }[/math] associated with a sequence [math]\displaystyle{ \{p_n(x)\} }[/math] of polynomials orthogonal with respect to a density [math]\displaystyle{ \rho(x) }[/math] are defined by

[math]\displaystyle{ q_n(x) = \int_\mathbb{R}\! \frac{p_n(t) - p_n(x)}{t - x} \rho(t)\,dt. }[/math]

To see that the functions [math]\displaystyle{ q_n(x) }[/math] are indeed polynomials, consider the simple example of [math]\displaystyle{ p_0(x)=x^3. }[/math] Then,

[math]\displaystyle{ \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} }[/math]

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

See also

• Secondary measure