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Analytic noncontinuability of certain classes of power series. (English. Russian original) Zbl 0972.30002
Math. Notes 64, No. 3, 410-413 (1998); translation from Mat. Zametki 64, No. 3, 474-477 (1998).
The author studies general classes of lacunary power series. Fix $$\alpha>0$$ and $$(m_{s}) _{s=1} ^\infty \subset\mathbb{N}$$. Let $$\beta_{s}$$ denote the fractional part of $$\alpha m_{s}$$. Assume that $$\lim_{N \to+ \infty} (1/N)\# \{s\in{\mathbf N}: m_{s}\leq N\}>0$$ and the sequence $$(\beta _{s}) _{{s}=1} ^\infty$$ is uniformly distributed in $$[0,1]$$. Let $$a_{s}: = f(\beta _{s})$$, $$s\in\mathbb{N}$$, where $$f$$ is a Riemann-integrable function on $$[0,1]$$, whose Fourier coefficients can vanish only for a finite number of indices. Then the unit disc is the domain of existence of the function $$F(z)= \sum_{{s}1} ^\infty a_{s} z^{m_{s}}$$. The same remains true if $$a_{s}:= \sum_{\nu=0}^{k} m_{s}^\nu p_\nu (\beta_{s})$$, $$s\in \mathbb{N}$$, where $$p_\nu$$ is a nonconstant polynomial, $$\nu=0,\cdots,k$$.
##### MSC:
 30B40 Analytic continuation of functions of one complex variable 11J71 Distribution modulo one
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##### References:
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