Secondary polynomials

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

Secondary measure

Every Eisenstein integer a + bω whose norm a2 − ab + b2 is a rational prime is an Eisenstein prime. In fact, every Eisenstein prime is of this form, or is a product of a unit and a rational prime congruent to 2 mod 3.

This is a discrete Fourier transform.

Mathematics Encyclopedia