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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...

Giuseppe Negro
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ghthorpe
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    Isn't it easier to use the (provable) fact that the functions $H_n(x)e^{-x^2/2}$ are obtained from $e^{-x^2/2}$ by "ladder operators" (meaning application of ${d\over dx}-x$)? This certainly gives Rodriguez' formula, and I think also easily gives the generating function as a corollary. Is there a reason you don't want to do it this way? – paul garrett Jun 10 '18 at 22:15
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    Here https://math.stackexchange.com/a/2747248/82615 is an induction method. Here https://math.stackexchange.com/q/679970/82615 is a direct method, using the 'physicist' definition here https://en.wikipedia.org/wiki/Hermite_polynomials#Definition . This direction arises in solving the Harmonic oscillator with creation/annihilation operators, leaving one with $(\xi - \frac{d}{d \xi})^n$ and needing to show it is equivalent to Hermite polynomials, c.f. Kaku Superstrings Sec. 1.8. – bolbteppa Jun 13 '18 at 05:38

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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.

user_phys
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