Let $\{m_i\}_{i=0}^\infty$ be a subsequence of nature number sequence and then consider the following function: $$f(z)=\sum_{i=0}^\infty z^{m_i}, |z|<1,$$ which is an analytical function on $|z|<1$. For example, $m_i=2i$, gives $f(z)=1+z^2+z^4+...=\tfrac{1}{1-z^2},$ which can be continued to be a meromorphic function on $\mathbb{C}$. Does this analytical continuation valid for all subsequence ${m_i}$ ? In other words, for any given subsequence $\{m_i\}$, can $f(z)$ be continued to be a meromorphic function on $\mathbb{C}$?
As noted by Martin, there are counterexamples (lacunary functions) for example $m_i = 2^i$. Hence I add a condition on $\{m_i\}$: $$\limsup_{i\to\infty} \tfrac{m_{i+1}}{i+1} < \infty.$$