2022/09/18 by Mateusz Kwaśnicki, Kwaśnicki, Mateusz, Jacek Wszoła +1
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2209.08641
openalex publication_date 2022/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A nonnegative real function f is said to be bell-shaped if it converges to zero at ±∞ and the nth derivative of f changes sign n times for every n = 0, 1, 2, … In a similar way, we may say that a nonnegative sequence ak is bell-shaped if it converges to zero and the nth iterated difference of ak changes sign n times for every n = 0, 1, 2, … Bell-shaped functions were recently characterised by Thomas Simon and the first author. In the present paper we provide an analogous description of bell-shaped sequences. More precisely, we identify 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.