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A connection between power series and Dirichlet series. (English) Zbl 07267866
In the paper the authors investigate relations between the analytic continuation of the Taylor series $$H(z)=\sum _{n=0}^{\infty} h_n z^n$$ to the boundary point of its disk of convergence and the analytic continuation of the corresponding Dirichlet series $$D(s)=\sum_{n=1}^{\infty}\frac{h_{n-1}}{n^s}$$ beyond its half-plane of convergence.
First, the authors obtain the following interpolation result for power series coefficients: if $$f(z)=\sum_{n=-k}^{\infty} a_n z^n, \ k\geq 0$$, is a convergent in $$\{z:\ 0<|z|<R\}$$ Laurent series, then for any fixed $$r\in (0, R)$$ the analytic continuation of the integral $$F(s)=\frac{1}{\Gamma (s)} \int_{0}^{r} f(t) t^{s-1} dt$$, is a meromorphic function on $$\mathbb{C}$$ whose singularities are a simple pole at $$s=k$$ and possible simple poles at $$s=1,2,\ldots, k-1$$, with residues $$Res(F; n)=\frac{a_{-n}}{(n-1)!}$$, and satisfying $$F(-n)=(-1)^n n! a_n, \ n=0, 1, 2, \ldots$$
Then they prove the main result of the paper: if a power series $$H(z)=\sum _{n=0}^{\infty} h_n z^n$$ converges inside the unit disc and has a meromorphic extension to a function with a pole at $$z=1$$, then the Dirichlet series $$D(s)=\sum _{n=1}^{\infty}\frac{ h_{n-1}} {n^s}$$ has a meromorphic extension to $$\mathbb{C}$$. For this purpose they apply their interpolation result to a function $$f(t)=e^{-t}H(e^{-t})$$. Several interesting counterexamples, illustrated difficulties of finding a converse to the main theorem, are considered. Finally, the authors extend their results to power series with a non-isolated singularity at $$z=1$$.
##### MSC:
 30B10 Power series (including lacunary series) in one complex variable 30B40 Analytic continuation of functions of one complex variable 30B50 Dirichlet series, exponential series and other series in one complex variable 30D30 Meromorphic functions of one complex variable, general theory
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