Secondary polynomials explained

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 byq _{n(x)} = \int _{R} (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 ^{3 .} _{0 (x) = x} Then,\begin{align} q _{0(x)} &{} = \int _{R} (t ^{3} - x ^{3})/(t - x) \rho(t)dt \\ &{} = \int _{R} ((t - x)(t ^{2+tx+x} ^{2)})/(t - x) \rho(t)dt \\ &{} = \int _{R} (t ^{2+tx+x} ^{2)\rho(t)dt} \\ &{} = \int _{R} t ^{2\rho(t)dt +} x\int _{R} t\rho(t)dt + x ^{2 \int} _{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

• Secondary measure