Two-sided bell-shaped sequences

Abstract A nonnegative real function f is bell-shaped if it converges to zero at $$\pm \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and the n th derivative of f changes sign n times for every $$n = 0, 1, 2, \ldots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> </mml:mrow> </mml:math> Similarly, a nonnegative sequence $$(a(k): k \in \mathbb {Z})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is bell-shaped if it converges to zero at $$\pm \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and the n th iterated difference of a ( k ) changes sign n times for every $$n = 0, 1, 2, \ldots $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> </mml:mrow> </mml:math> 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 Pólya 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.