Given a dot product on functions of the form $\langle f, g \rangle = \iint_{\mathbb{R}} w(x)f(x)g(x)\, dx$, we can consider orthogonal polynomials, i.e. of the form $\langle f, g\rangle = 0$. A question is if we can generate sequences $p_n$ of these polynomials of consecutive degree -- such a set would then be a basis for polynomials up to that degree (do you see why?).
What we want is to pin down what $p_n$ must be given $p_1,\dots p_{n-1}$. What this means is trying to express the "non-leading part" of $p_n$ in terms of these former terms. We can access a "non-leading part" of the polynomial by normalizing the polynomials to monic and considering the $n-1$-degree polynomial $p_n-xp_{n-1}$. What are the components of this guy in the basis of $p_1,\dots p_{n-1}$? Well, writing
\[{p_n} - x{p_{n - 1}} = \sum\limits_{m < n} {{a_m}{p_m}} \]
Then for all $m<n$,
\[\begin{align}
{a_m} &= \frac{{\left\langle {{p_n} - x{p_{n - 1}},{p_m}} \right\rangle }}{{\left\langle {{p_m},{p_m}} \right\rangle }} \\
&= - \frac{{\left\langle {x{p_{n - 1}},{p_m}} \right\rangle }}{{\left\langle {{p_m},{p_m}} \right\rangle }} \\
&= - \frac{{\left\langle {{p_{n - 1}},x{p_m}} \right\rangle }}{{\left\langle {{p_m},{p_m}} \right\rangle }} \\
\end{align} \]
Now, $xp_m$ has degree $m+1$. So if $m+1<n-1$, ${\left\langle {{p_{n - 1}},x{p_m}} \right\rangle }=0$. So the only $m$s that we need to bother about are $n-2$ and $n-1$. Therefore:
\[{p_n} - x{p_{n - 1}} = {a_{n - 2}}{p_{n - 2}} + {a_{n - 1}}{p_{n - 1}}\]
So:
\[\begin{align}
{p_n} &= - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 2}}} \right\rangle }}{{\left\langle {{p_{n - 2}},{p_{n - 2}}} \right\rangle }}{p_{n - 2}} + \left[ {x - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}} \right]{p_{n - 1}} \\
&= - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 2}} - {p_{n - 1}}} \right\rangle + \left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 2}},{p_{n - 2}}} \right\rangle }}{p_{n - 2}} + \left[ {x - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}} \right]{p_{n - 1}} \\
&= - \frac{{\left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 2}},{p_{n - 2}}} \right\rangle }}{p_{n - 2}} + \left[ {x - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}} \right]{p_{n - 1}}
\end{align} \]
Examples:
- Legendre polynomials: $w$ is the indicator for $[-1,1]$. Sequence: $1, x, x^2-\frac13, x^3-\frac35x,\dots$
- Chebyshev polynomials: $w$ is $(1-x^2)^{-1/2}$ on $[-1,1]$. Sequence: $\cos(n\arccos x)$
- Laguere polynomials: $w$ is $e^{-x}$ on $\mathbb{R}^{\ge 0}$. Sequence: $1, x-1, x^2-4x+2, x^3-9x^2+18x-6$
- Hermite polynomials: $w$ is $e^{-x^2}$ everywhere. Sequence: $1, x, x^2-\frac12, x^3-\frac32 x$
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