Proving orthogonality of a family of polynomials with respect to inner product

Question

there's a question that i've been stuck on and would really like some help with.

Consider a space of polynomials defined on the real line with the following inner product $$ \langle p,q\rangle = \int_{\mathbb R} p(x)e^{-x^2/2}q(x) dx $$ where p(x) and q(x) are polynomials.

Consider a family of polynomials defined recursively over $\mathbb R$ $$H_0(x) = 1$$ $$H_1(x) = x$$ $$H_{n+1}(x)=xH_n(x) - H'_n (x) $$

So the question asks us to show that these polynomials form an orthogonal family with respect to the inner product that was previously defined.

Any help would be greatly appreciated!

Answer 1

We will use induction, but first we make some observations.

First, observe that for $\phi(x)=-e^{-x^2/2}\,f(x)$, it holds that $\phi'(x)=e^{-x^2/2}\,\big(xf(x)-f'(x)\big)$. In particular, with $\phi_n(x)=e^{-x^2/2}H_n(x)$ we have $\phi_n'(x)=-\phi_{n+1}(x)$.

Secondly, it is a straightforward exercise to check that $H_n$ is a monic $n$-degree polynomial for all $n$.

Finally, it is a common exercise to show that $\lim_{x\to\pm\infty}e^{-x^2/2}\,p(x)=0$ whenever $p$ is a polynomial.

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Now, onto the induction. The base case uses $p=H_0$ and $q=H_1$, which should be easy enough to check.

For the induction step, suppose that for all $j \in \{1,\dots,n\}$ and all $i$ with $0\leq i < j$ we have $\langle H_i,H_j\rangle = 0$. We will show the same holds true when $j=n+1$. We have:

\begin{align} \langle H_i,H_{n+1}\rangle=\int\,H_i\,H_{n+1}\,e^{-x^2/2}\,dx &= \int\,H_i\,\phi_{n+1}(x)\,dx\\ &= \int\,H_i\,\Big(-\phi_n'(x)\Big)\,dx\tag{1}\\ &= \Big[H_i\,\big(-\phi_n\big)\Big]-\int\,H_i'\,\Big(-\phi_n(x)\Big)\,dx\tag{2}\\ &= \underbrace{\Big[-H_i\phi_n\Big]}_{(*)}+\int\,H_i'\,\phi_n(x)\,dx\\ \end{align}

In $(1)$ we used our first observation. In $(2)$ we used integration by parts. The boundary component $(*)$ is $0$ as per our third observation above.

In other words, we have

$$\int\,H_i\phi_{n+1}(x)\,dx=\int\,H_i'\,\phi_n(x)\,dx$$

We can repeat the process, at each step differentiating $H_i$ one additional time and reducing the index of $\phi_k$ by $1$. Since $i<n+1$ as per our induction hypothesis, in $i$ steps the $H_i$ factor of the ingrand will be reduced to the constant $i!$ (here, we have used our second observation) and the index on what remains of $phi_k$ will be positive. At this point, the integral will have been reduced to

$$i!\int\,\phi_{n+1-i}(x)\,dx= i!\int\, H_{n+1-i}(x)\,e^{-x^2/2}\,dx=i!\,\langle H_{n+1-i},H_0\rangle,$$

which is $0$ as per our induction hypothesis.

You may notice the argument above doesn't quite work when $i=0$... In this case, however, the problem would have been solved earlier, in equation $(1)$: here we could apply the Fundamental Theorem of Calculus and our third observation to conclude that the integral vanishes.