Fabry gap theorem

In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them.

Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence.

Let (αj)j∈N be a sequence of complex numbers such that the power series has radius of convergence 1.

Then the unit circle is a natural boundary for the series f. A converse to the theorem was established by George Pólya.

If lim inf pn/n is finite then there exists a power series with exponent sequence pn, radius of convergence equal to 1, but for which the unit circle is not a natural boundary.