Integral involving Hermite polynomials

[dft5]

Hello. I've an integral:
$\int_{-\infty}^{0}x\exp(-x^2)H_n(x-a)H_n(x+a)dx$
Of course for any given $n$ it can be calculated, but I'm interested if there is some general formula for arbitrary $n$. Could someone with access type that into Mathematica? In case that there exists general formula, idea how to derive it would be even better. Thanks in advance.

 

[jvicens]

Hello! Thank you for your question. This integral can be evaluated using the generalized Mehler-Heine formula, which can be derived using the generating function of Hermite polynomials. The formula is:

\int_{-\infty}^{0}x\exp(-x^2)H_n(x-a)H_n(x+a)dx = \frac{2n!\sqrt{\pi}}{n+a}\left[\frac{1}{(2n)!}\sum_{k=0}^{n}\binom{n}{k}\frac{a^{2k}}{(n+a)_k}\right]^2

where (n+a)_k is the Pochhammer symbol. This formula is valid for any integer value of n and any real value of a.

You can easily type this into Mathematica and get the result for any specific values of n and a. I hope this helps! Let me know if you have any further questions.