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


Author: Torben Koch
Journal: Proc. Amer. Math. Soc. 148 (2020), 2149-2155
MSC (2010): Primary 33C10, 26D07; Secondary 44A10, 60G40, 60J60
DOI: https://doi.org/10.1090/proc/14896
Published electronically: January 28, 2020
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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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Additional Information

Torben Koch
Affiliation: Center for Mathematical Economics (IMW), Bielefeld University, Universitätsstrasse 25, 33615, Bielefeld, Germany
Email: t.koch@uni-bielefeld.de

DOI: https://doi.org/10.1090/proc/14896
Keywords: Hermite functions, parabolic cylinder functions, Tur\'an-type inequalities, Ornstein-Uhlenbeck process.
Received by editor(s): June 5, 2019
Received by editor(s) in revised form: October 4, 2019
Published electronically: January 28, 2020
Additional Notes: The author was supported by the German Research Foundation (DFG) through the Collaborative Research Centre 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications”.
Communicated by: Yuan Xu
Article copyright: © Copyright 2020 American Mathematical Society