Secondary polynomials

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 ho(x) are defined by

: q_n(x) = int_mathbb{R}! frac{p_n(t) - p_n(x)}{t - x} ho(t),dt.

To see that the functions q_n(x) are indeed polynomials, consider the simple example of p_0(x)=x^3. Then,

:egin{align} q_0(x) &{}= int_mathbb{R} ! frac{t^3 - x^3}{t - x} ho(t),dt \&{}= int_mathbb{R} ! frac{(t - x)(t^2+tx+x^2)}{t - x} ho(t),dt \&{}= int_mathbb{R} ! (t^2+tx+x^2) ho(t),dt \&{}= int_mathbb{R} ! t^2 ho(t),dt + xint_mathbb{R} ! t ho(t),dt + x^2int_mathbb{R} ! ho(t),dt end{align}

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

See also

* Secondary measure


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