Asymptotics of Hermite and hypergeometric function

Question

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 the 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 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?

Answer 1

I suppose I should be more precise than what I wrote in my comment: as already mentioned, whenever one of the "numerator parameters" of a hypergeometric function is a negative integer -m, the series terminates (the Pochhammer symbols in the terms of degree higher than m vanish), i.e. your hypergeometric function becomes a polynomial.

Using this, the degree m term of the polynomial is

$\frac{(-m)_m (m)_m}{\left(\frac1{2}-m\right)_m}\frac{\beta^m}{m!}$

or, using the properties of the Pochhammer symbols and the factorial:

$2^{2m-1}\beta^m$

Replacing $\beta$ with $(2-\alpha^{-2})^{-1}$ and multiplying by the factors in front of the hypergeometric expression nets you

$\frac{(-1)^m 2^{3m-\frac{3}{2}}\Gamma\left(\frac1{2}+m\right)}{\alpha}$

What complicates things, however, are the factors in front of the hypergeometric function, which when expanded is a polynomial in odd powers of the variable $\alpha^{-1}$.

It's a bit late here, so I suppose I'll let someone else finish the analysis...

Answer 2

For the asymptotic behavior of the hypergeometric function your stated, I think you would be referred to [Farid Khwaja, S.; Olde Daalhuis, A. B. Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (2014), no. 6, 667–710.], or to the similar papers by the authors.