# Higher derivatives of an exponential function

Let $$p_n(x)e^{-x^2}$$ be the $n$th derivative of $$e^{-x^2}.$$ Find a formula for $p_n(x)$. We have $p_1(x)=-2x, p_2(x)=4x^2-2$, etc. But what is the general formula for $p_n$?

• – Aryabhata Nov 26 '10 at 3:05

$p_n$ satisfies the recurrence, $p_{n+1}(x) = p_n'(x) - 2xp_n(x)$ with $p_0(x) = 1$. This looks to me like it might give you some Tchebyshev polynomial. Something like $p_n(x) = 2 (-1)^n T_n(x)$ .