Notes on Hermite Polynomials

Question

I am currently studying Hermite Polynomials, however every book that I have read (Mathematical Methods for Physicists-Arfken,Mathematical methods for physics and engineering-Cambridge University Press, Methods of Mathematical Physics - Hilbert) only make a relatively brief reference to them. I have searched over the internet for notes that not only display, but also prove the properties of Hermite Polynomials plus some solved practice problems but I haven't found anything that helped me. Is there any reference that you can give me? BTW I am a physics student.

Answer 1

Consider $p(x) = a x^2 + b x + c$ a quadratic polynomial with $a>0$. One checks easily that for every $n\ge 0$ $$H_n(x) \colon = (e^{-p(x)})^{(n)} \cdot e^{p(x)}$$ is a polynomial of degree $n$. Moreover, we have the Taylor expansion

$$\sum_{n=0}^{\infty} \frac{(e^{-p(x)})^{(n)}}{n!}t^n = e^{-p(x+t)}$$ and so

$$\sum_{n=0}^{\infty} \frac{H_n(x)}{n!} t^n = e^{-p(x+t)+p(x)}$$

the exponential generating function of the sequence $(H_n(x))$.

To show the orthogonality with respect to the weight $e^{-p(x)}$ we'll use the method indicated by @reuns: in the comments above. So consider

$$f(t, x) \colon = \sum_{n=0}^{\infty} \frac{t^n}{n!} H_n(x) e^{-\frac{p(x)}{2}}$$

We want to show that

$$\int_{-\infty}^{\infty} f(t,x)\cdot f(x,s) d x$$

is a function of $s t$ ( that is, the exponents of all the nonzero terms in $s$ and $t$ are equal. Now the above integral equals

$$\int_{-\infty}^{\infty}e^{-p(t+x) + p(x) - p(x+s)} dx = \sqrt{\frac{\pi}{a}} e^{ \frac{b^2- 4 a c}{4 a}}\cdot e^{2 a s t}$$

From the above we see that the $H_n(x)$ are orthogonal, and also we find their squared norm.

The last formula follow from the following:

For a polynomial $p(x)= (a x^2 + b x + c)$, with $a> 0$, we have $p(x) = a(x+\frac{b}{2a})^2 + \frac{-(b^2- 4 a c)}{4 a}$. Since $\int_{\mathbb{R}} e^{-a x^2}= \sqrt{\frac{\pi}{a}}$, we get $\int_{\mathbb{R}} e^{-p(x)} = \sqrt{\frac{\pi}{a}} \cdot e^{\frac{\Delta}{4 a}}$. Now check that the leading term of $p_1(x) = p(t+x) + p(s+x) - p(x)$ is still $a$, while the discriminant is $\Delta_1 = \Delta + 8 a^2 s t$.