ca.analysis and odes - Asymptotics of Hermite and hypergeometric function

I am looking for the asymptotics of the following integral

$int_{mathbb{R}} H_m^2(x) {rm e}^{-2 alpha^2 x^2} {rm d} x = 2^{m-1/2} alpha^{-2m -1} (1-2alpha^2)^m Gamma(m+1/2) ~ _2F_1left(-m,m,1/2-m,frac{alpha^2}{2alpha^2-1}right)$

where $H_m$ is the $m^{rm th}$ Hermite polynomial (orthogonal under the weight ${rm e}^{-x^2}$), and $_2F_1$ is the hypergeometric function.

I found this formula from p. 803 of "Table of Integrals, Series, and Products" by Gradshteyn-Ryzhik. However, I have idea about the asymptotics of the $_2F_1$ term. Can anyone enlighten me on the asymptotics of

$_2F_1left(-m,m,1/2-m,betaright)$

when $m$ is large? In fact I tried mathematica and it seems $_2F_1left(-m,m,1/2-m,betaright) sim |4 beta|^m$. Any reference on this issue?

Now given the above asymptotics is true, observe that the norm of $H_m$ under the weight ${rm e}^{-2 alpha^2 x^2}$ has the same exponent for all $alpha$, including the original weight ($alpha^2 = 1/2$). Is this a common phenomenon for orthogonal polynomials?

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