View source for Lacunary power series

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A series

\begin{equation}\label{lac} f(z)=\sum_{k=1}^\infty a_kz^{\lambda_k}\end{equation}

with gaps (lacunas), so that the exponents $\lambda_1,\lambda_2,\ldots,$ do not run through all the natural numbers. Depending on the properties of the sequence $\{\lambda_k\}$ one obtains many properties of the series \ref{lac}. Thus, if

$$\lambda_{k+1}-\lambda_k>\theta\lambda_k,\quad k=0,1,\ldots,\quad \theta>0,$$

and the series \ref{lac} converges in the disc $\lvert z\rvert<R$, $0<R<\infty$, then all points of the circle $\lvert z\rvert=R$ are singular for $f(z)$ ([[Hadamard theorem#Hadamard's gap theorem|Hadamard's gap theorem]]). A strengthening of this theorem is Fabry's gap theorem ([[Fabry theorem|Fabry theorem]]). If the lower density

$$\liminf_{k\to\infty}\frac{k}{\lambda_k}=0,$$

then $f(z)$ is a single-valued analytic function with simply-connected domain of existence (Pólya's theorem). See also [[Over-convergence|Over-convergence]].

====References====
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|valign="top"|{{Ref|Bi}}||valign="top"| L. Bieberbach,   "Analytische Fortsetzung" , Springer  (1955)  pp. Sect. 3
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|valign="top"|{{Ref|Di}}||valign="top"|  P. Dienes,   "The Taylor series" , Oxford Univ. Press &amp; Dover  (1957)
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|valign="top"|{{Ref|Ti}}||valign="top"| E.C. Titchmarsh,   "The theory of functions" , Oxford Univ. Press  (1979)
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