2019/05/24 by T. L. Koch, Torben Koch, Koch, Torben
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Statistical Mechanics and Entropy #math.CA
paper · pdf · doi:10.48550/arxiv.1905.10274
arxiv created 2019/05/24 · openalex publication_date 2019/05/24 · arxiv updated 2019/05/27 · openalex created_date 2022/07/23 · openalex updated_date 2026/07/28
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 to 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.