Hello,
is there any way of calculating the following integral in closed form:
$\displaystyle \int_0^1{\mbox{H}_n(x) e^{-\frac{x^2}{2}}\mbox{d}x}$?
Thank you
What if you plugged the Rodriguez formula into the integral? You'd get
$\displaystyle \displaystyle{\int_{0}^{1}H_{n}(x)\,e^{-\frac{x^{2}}{2}}\,dx=
\int_{0}^{1} (-1)^n e^{\frac{x^{2}}{2}} \frac{d^n}{dx^n} \left( e^{-x^2/2}\right)\,e^{-\frac{x^{2}}{2}}\,dx=(-1)^{n}\int_{0}^{1} \frac{d^n}{dx^n} \left( e^{-x^2/2}\right)\,dx.}$
Can you see where to go from here?
The plot thickens... It seem that "my" Hermite polynomials are the physicist's version and have exp(-x^2) in the definition so the procedure wouldn't work that trivially. But nevertheless, I managed to evaluate the integral using recurrence relations. Thanks to the both of you!