integration involving Hermit function

Question

Iam trying to evaluate the integration of the following two products .. every one by itself

where Hn-1 is the Hermite function . All my tries ends with a zero value for the two integrations, but it must not . Any Help ?

Answer 1

Hint. $\forall n \neq k$ $$\langle H_{n}, H_{k} \rangle = \int_{-\infty}^{+\infty}H_{n}(x)H_{k}(x)e^{-x^{2}/2}dx$$ is identically zero implying that $H_{n}$ and $H_{k}$ are orthogonal. A lemma naturally follows;

Lemma 1. $$\int H_{n+1}e^{-x^{2}/2}dx=C-H_{n}e^{-x^{2}/2}$$ Where $C$ is a constannt. I suspect these relations will help you find your answer, by the weay my opinion is that I'm not surpised you're finding zeroes to the answer.

Edit Based on the requirement in the comment to this post, I will post a few observations, I will not go into the inductive proof.

Note that \begin{eqnarray} \int x e^{-x^{2}/2}dx &=& -e^{-x^{2}/2}+C \\ \int (x^{2}-1)e^{-x^{2}/2}dx &=& -xe^{-x^{2}/2}+C \\ \int (x^{3}-3x)e^{-x^{2}/2}dx &=& -(x^{2}-1)e^{-x^{2}/2}+c \end{eqnarray}

Further Edit Also note that for general $n$ $$x^{n}=H_{n}(x)+a_{n-1}H_{n-1}(x)+a_{n-2}H_{n-2}(x)+ \ldots a_{1}H_{1}(x)+a_{0}H_{0}(x)$$