论文标题
Bernoulli操作员和Dirichlet系列
Bernoulli Operators and Dirichlet Series
论文作者
论文摘要
我们介绍并研究了一些称为Bernoulli操作员的（无限级）离散衍生运算符。它们与一类Power系列（TAME Power Series）相关联，其中包括在单元磁盘中收敛的功率系列，最多在$ z = 1 $的情况下具有极点奇异性，并且对以$ z = 1 $为中心的单位磁盘具有分析性延续，并具有可能的Mittag-Leffler Type的孤立奇异性。我们表明，它们都自然地作用于函数$ f（s，t）$的矢量空间，并在拉普拉斯 - 梅尔蛋白变换的图像中具有（单个值）分析性延续到复杂平面，并具有可能的孤立奇异性。对于$ s $，在某个正确的半平面中，伯努利操作员的动作由Dirichlet-type系列给出，因此，此类系列将分析延续到复杂的平面上，并允许对奇异性进行精确描述。对于$ f（s，t）= t^s $的特定情况，伯努利操作员的动作提供了与驯服功率系列相关的Dirichlet系列的分析性延续。在这种情况下，我们记录有关电线杆位置，其居住和特殊值的详细信息，并证明了具有指定的杆，残基和特殊值的Tame Dirichlet系列的独特性。
We introduce and study some (infinite order) discrete derivative operators called Bernoulli operators. They are associated to a class of power series (tame power series), which include power series that converge in the unit disk, have at most a pole singularity at $z=1$, and have analytic continuation to the unit disk centered at $z=1$ with possible isolated singularities of Mittag-Leffler type. We show that they all naturally act on, and take values into, the vector space of functions $f(s,t)$ in the image of the Laplace-Mellin transform that have (single valued) analytic continuation to the complex plane with possible isolated singularities. For $s$ in some right half-plane the action of the Bernoulli operator is given by a Dirichlet-type series and, as a consequence, such series acquire analytic continuation to the complex plane and allow a precise description of the singularities. For the particular case of $f(s,t)=t^s$, the action of the Bernoulli operators provide the analytic continuation of the Dirichlet series associated to tame power series. In this case, we record detailed information about the location of poles, their resides, and special values, as well as prove the uniqueness of tame Dirichlet series with specified poles, residues, and special values.
