New answers tagged hermite-polynomials
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Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
Your conjecture is not true. Moreover,
$$J_m(y):=\int_y^\infty dx\, e^{-x^2}H_m(x)^2\sim c_m:=\frac12\,\pi^{1/2}\,2^m m! \tag{1}\label{1}$$
whenever
$$0\le y=o(m^{1/2}).$$
Indeed, recalling that $H_m^...
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Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
Let me first consider the case $x_0=0$, when the integral has a closed form expression:
$$\int^{\infty}_{0}|H_m(x)|^2 e^{-x^2}\,dx=\sqrt{\pi } \,2^{m-1} m!\;\;(m\in\mathbb{N}).$$
See, for example, ...
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