Bell-shaped sequences

A nonnegative real function $f$ is said to be bell-shaped if it converges to zero at $\pm \mathrm{\infty }$ and the $n$th derivative of $f$ changes sign $n$ times for every $n=0,1,2,\dots$ In a similar way, we may say that a nonnegative sequence $a_k$ is bell-shaped if it converges to zero and the $n$th iterated difference of $a_k$ changes sign $n$ times for every $n=0,1,2,\dots$ 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.