en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-26T20:48:36Z

Fix CS1 deprecated date parameter errors

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{{unreferenced|date=August 2012}}
In mathematics, the '''secondary polynomials''' <math>\{q_n(x)\}</math> associated with a [[sequence]] <math>\{p_n(x)\}</math> of [[polynomials]] [[orthogonal polynomials|orthogonal]] with respect to a density <math>\rho(x)</math> are defined by

: <math> 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>q_n(x)</math> are indeed polynomials, consider the simple example of <math>p_0(x)=x^3.</math> Then,

:<math>\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>x</math> provided that the three integrals in <math>t</math> (the [[Moment (mathematics)|moments]] of the density <math>\rho</math>) are convergent.

==See also==

* [[Secondary measure]]

[[Category:Polynomials]]
{{math-stub}}

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en>Monkbot: Fix CS1 deprecated date parameter errors