Universal bounds and monotonicity properties of ratios of Hermite and parabolic cylinder functions

Koch, Torben

doi:10.1090/proc/14896

Universal bounds and monotonicity properties of ratios of Hermite and parabolic cylinder functions
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Proc. Amer. Math. Soc. 148 (2020), 2149-2155 Request permission

Abstract:

We obtain so far unproved properties of a ratio involving a class of Hermite and parabolic cylinder functions. Those ratios are shown to be strictly decreasing and bounded by universal constants. Differently from the usual analytic approaches, we employ simple purely probabilistic arguments to derive our results. In particular, we exploit the relation between Hermite and parabolic cylinder functions and the eigenfunctions of the infinitesimal generator of the Ornstein-Uhlenbeck process. As a byproduct, we obtain Turán-type inequalities.

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