The Hermite polynomials I am using satisfy the recurrence $H_k'(x) = kH_{k-1}(x)$ and they satisfy the property if $d\mu_{1/2} = (2\pi)^{-1/2}e^{-x^2/2}$ is the standard Gaussian then $||H_k||_{L^2(\mu_{1/2})}^2 = k!$. For instance, $H_2(x) = x^2 - 1$, and $H_3(x) = x^3 - 3x$, and $H_4(x) = x^4 - 6x^2 + 3$.
Is there a simple formula for obtaining the norm of the same Hermite polynomials but wrt another Gaussian measure? That is, if $d\mu_\alpha = (4\pi \alpha)^{-1/2}e^{-\alpha x^2}$ is another normalized Gaussian, is there a simple formula for $||H_k||_{L^2(\mu_\alpha)}^2$ in terms of $\alpha$ and $k$? Or are there lower and upper bounds?
Even for $\alpha = 1$ the only way I see I can move forward with this calculation is making the substitution $x = y/\sqrt{2}$ so that $||H_k||_{L^2(\mu_\alpha)}^2 = \int_{\mathbb R} H_k(x)^2d\mu_1(x) = \frac{1}{2}\int_{\mathbb R} H_k(y/\sqrt{2})^2d\mu_{1/2}(y)$ is the norm of a scaled Hermite polynomial with respect to the standard Gaussian. But I don't know any ways to control this integral.