Bell-shaped sequences

TL;DR

This paper characterizes bell-shaped sequences, which are sequences with sign-changing difference properties, by linking them to convolutions of Pólya frequency and completely monotone sequences, and describes their generating functions via Pick functions.

Contribution

It provides an analogous characterization of bell-shaped sequences, extending recent work on bell-shaped functions, and links these sequences to specific classes of generating functions.

Findings

Bell-shaped sequences are characterized as convolutions of Pólya frequency and completely monotone sequences.

The generating functions of bell-shaped sequences are identified as exponentials of Pick functions.

The paper extends the theory of bell-shaped functions to discrete sequences.

Abstract

A nonnegative real function $f$ is said to be bell-shaped if it converges to zero at $\pm\infty$ and the $n$th 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 $n$th 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 bell-shaped sequences. More precisely, we identify bell-shaped sequences with convolutions of P\'olya frequency sequences and completely monotone sequences, and we characterise the corresponding generating functions as exponentials of appropriate Pick functions.