Bell-shaped sequences
Volume 271 / 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.