Year: 2025
Type: article
Abstract: A nonnegative real function f is bell-shaped if it converges to zero at $$\pm \infty $$ ± ∞ and the nth derivative of f changes sign n times for every $$n = 0, 1, 2, \ldots $$ n = 0 , 1 , 2 , … Similarly, a nonnegative sequence $$(a(k): k \in \mathbb {Z})$$ ( a ( k ) : k ∈ Z ) is bell-shaped if it converges to zero at $$\pm \infty $$ ± ∞ and the nth iterated difference of a(k) changes sign n times for every $$n = 0, 1, 2, \ldots $$ n = 0 , 1 , 2 , … A characterisation of bell-shaped functions was given by Thomas Simon and the first named author, and recently a similar result for one-sided bell-shaped sequences was found by the authors. In the present article we give a complete description of two-sided bell-shaped sequences. Our main result proves that bell-shaped sequences are convolutions of Pólya frequency sequences and what we call absolutely monotone-then-completely monotone sequences, and it provides an equivalent, and relatively easy to verify, condition in terms of holomorphic extensions of the generating function. We also prove that if f is a bell-shaped function, then f(k) is a bell-shaped sequence.
Source: Mathematische Zeitschrift
Authors: Mateusz Kwaśnicki, Jacek Wszoła
Institution: Wrocław University of Science and Technology
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4414315532