There are a variety of ways of first defining the Hermite Polynomials in a certain way and then to derive alternative representations of them. For example in Mary Boas' Mathemmatical methods (p. 607, 3rd edition) she starts with the differential equation $y''_n-x^2y_n = -(2n+1)y_n$. Arfken's Mathematical Methods (p. 817, 6th edition) starts with a generating function $g(x,t)=e^{2xt-t^2}=\sum_{n=0}^{\infty}\frac{t^n}{n!}H_n(x)$. Griffiths Intro to Q.M. (p.57,problem 2.17, 2end edition) starts with Rodrigues formula: $H_n(x)= (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$.
I'd like to do the following: Start from the defintion $H_n(x)= e^{x^2/2}(x-\frac{d}{dx})^ne^{-x^2/2}$ and then derive both Rodrigues Formula $H_n(x)= (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$ and then the generating function $g(x,t)=e^{2xt-t^2}=\sum_{n=0}^{\infty}\frac{t^n}{n!}H_n(x)$.
I'm not sure where to begin. My primary interest in this is quantum mechanics and how this Hermite Polynomial is used as a solution to the quantum Harmonic oscillator. I have never used Hermite Polynomials (or Laguerre, Legendre or Bessel functions)… So I'd be grateful for some advice on how to do this.
edit: Okay so here's my approach for deriving Rodrigues Formula $H_n(x)= (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$: I use induction on $n$. For $n = 0$ both $H_n(x)= e^{x^2/2}(x-\frac{d}{dx})^ne^{-x^2/2}$ and Rodrigues give $1$. Now assume the equality holds for some $n$ and show it for $n+1$: but now I'm stuck again…
Best Answer
For a start, you can go from the Rodrigues Formula to the generating function by noting that the equality for the generating function holds if and only if $$\frac{d^n}{dt^n}|_{t=0} g(x,t) = H_n(x)$$ But the left hand side evaluates to $$\frac{d^n}{dt^n}|_{t=0} e^{2xt-t^2} = \frac{d^n}{dt^n}|_{t=0} e^{x^2} e^{-(x-t)^2} = e^{x^2}\frac{d^n}{dt^n}|_{t=0}e^{-(x-t)^2}$$ Now substitute $t=x-u$ in the expression on the right, so that it becomes $$e^{x^2}(-1)^n\frac{d^n}{du^n}|_{u=x}e^{-u^2}$$ which is exactly the Rodrigues formula.