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Mathematics > Classical Analysis and ODEs
Title: Two-sided bell-shaped sequences
(Submitted on 17 Apr 2024)
Abstract: A nonnegative real function f is bell-shaped if it converges to zero at plus and minus infinity and the nth derivative of f changes sign n times for every n = 0, 1, 2, ... Similarly, a two-sided nonnegative sequence a(k) is bell-shaped if it converges to zero at plus and minus infinity and the nth iterated difference of a(k) changes sign n times for every 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 P\'olya 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.
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