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Bell-shaped sequences

Volume 271 / 2023

Mateusz Kwaśnicki, Jacek Wszoła Studia Mathematica 271 (2023), 151-185 MSC: Primary 40A05; Secondary 26A51, 39A70, 60E10, 60E07. DOI: 10.4064/sm220923-2-2 Published online: 3 April 2023

Abstract

A nonnegative real function is said to be bell-shaped if it converges to zero at \pm \infty , and the nth derivative of f changes sign n times for every n = 0, 1, 2, \ldots In a similar way, we may say that a nonnegative sequence a_k is bell-shaped if it converges to zero, and the nth iterated difference of a_k changes sign n times for every n = 0, 1, 2, \ldots Bell-shaped functions were recently characterised by Thomas Simon and the first author. In the present paper we provide an analogous description of one-sided bell-shaped sequences. More precisely, we identify one-sided bell-shaped sequences with convolutions of Pólya frequency sequences and completely monotone sequences, and we characterise the corresponding generating functions as exponentials of appropriate Pick functions.

Authors

  • Mateusz KwaśnickiDepartment of Pure Mathematics
    Wrocław University of Science and Technology
    50-370 Wrocław, Poland
    e-mail
  • Jacek WszołaFaculty of Pure and Applied Mathematics
    Wrocław University of Science and Technology
    50-370 Wrocław, Poland
    e-mail

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