钟形序列
Bell-shaped sequences
摘要：
一个非负实函数$f$被称为钟形，如果它在$\pm\infty$处收敛到零，并且$f$的$n$阶导数对于每个$n = 0, 1, 2, \ldots$改变符号$n$次。我们可以说一个非负序列$a_k$是钟形的，如果它收敛到零，并且$a_k$的$n$次迭代差对于每个$n = 0, 1, 2, \ldots$ Bell改变符号$n$次。形函数最近由Thomas Simon和第一作者描述。在本文中，我们提供了一个类似的钟形序列的描述。更准确地说，我们确定钟形序列的卷积波利亚频率序列和完全单调序列，我们将相应的生成函数作为指数的适当的挑选功能。
英文摘要：
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 Pólya frequency sequences and completely monotone sequences, and we characterise the corresponding generating functions as exponentials of appropriate Pick functions.
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