Lacunary power series

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's gap theorem). A strengthening of this theorem is Fabry's gap 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.

References

[Bi]	L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3
[Di]	P. Dienes, "The Taylor series" , Oxford Univ. Press & Dover (1957)
[Ti]	E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979)