Two-sided bell-shaped sequences

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. We also prove that if f is a bell-shaped function, then f(k) is a bell-shaped sequence.