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I have a scaled probabilist's Hermite polynomial: \begin{equation} H_n(ax)e^{-bx^2} \end{equation} where $a$ and $b$ are both complex numbers. What is the asymptotic limit of this expression as $n \to \infty$? Thank you in advance!

user376231
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    Asymptotic expansion for physicists Hermite polynomials as $n\to\infty$: yikes – Aaron Hendrickson Mar 17 '21 at 01:37
  • Very useful. Thanks for sharing! – user376231 Mar 17 '21 at 01:54
  • This is not a simple problem. The large-$n$ asymptotics of the Hermite polynomial depends heavily on the location of $ax$ in the complex plane. You can re-express them in terms of parabolic cylinder functions (http://dlmf.nist.gov/18.15.E28) and apply the results of the paper F. W. J. Olver, Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders, J. Res. Nat. Bur. Standards Sect. B 63B (1959), pp. 131–169. – Gary Mar 17 '21 at 13:02
  • This is informative. Thanks! For now, I need the expansion at small $|x|$. The simplest form seems to works ok for the $a$ values I'm using. I might need different form at other parameter values. – user376231 Mar 19 '21 at 01:15

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