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I am looking for the asymptotics of the following integral

$\int_{\mathbb{R}} H_m^2(x) {\rm e}^{-2 \alpha^2 x^2} {\rm d} x = 2^{m-1/2} \alpha^{-2m -1} (1-2\alpha^2)^m \ \Gamma(m+1/2) ~ _2F_1\left(-m,m,1/2-m,\frac{\alpha^2}{2\alpha^2-1}\right)$

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 this issuethe asymptotics of

$_2F_1\left(-m,m,1/2-m,\beta\right)$

when $m$ is large? In fact I tried mathematica and it seems $_2F_1\left(-m,m,1/2-m,\beta\right) \sim |4 \beta|^m$. Any reference pleaseon this issue?

P.S. I suppose

Now given the integral above asymptotics is maximized at $\alpha^2 = 1/2$true, observe that is their original weight. When the weight is off, the energy should drop? So my guess is that norm of $H_m$ under the weight $_2F_1$ term is exponential and proportional to {\rm e}^{-2 \alpha^2 x^2}$has the same exponent for all$\frac{1}{(1-2\alpha^2)^m}$.alpha$, including the original weight ($\alpha^2 = 1/2$). Is this a common phenomenon for orthogonal polynomials?

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# Asymptotics of Hermite and hypergeometric function

I am looking for the asymptotics of the following integral

$\int_{\mathbb{R}} H_m^2(x) {\rm e}^{-2 \alpha^2 x^2} {\rm d} x = 2^{m-1/2} \alpha^{-2m -1} (1-2\alpha^2)^m \ \Gamma(m+1/2) ~ _2F_1\left(-m,m,1/2-m,\frac{\alpha^2}{2\alpha^2-1}\right)$

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 this issue? Any reference please?

P.S. I suppose the integral is maximized at $\alpha^2 = 1/2$, that is their original weight. When the weight is off, the energy should drop? So my guess is that the $_2F_1$ term is exponential and proportional to $\frac{1}{(1-2\alpha^2)^m}$.