Bernoulli Operators and Dirichlet Series

Abstract

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.