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there are examples of lacunary functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition.I want to know more examples of those functions,the more the better,especially the ones in closed form or series with integer coefficients.

And is there any research or theorem by which we can know or decide such a series that does not satisfy Fabry or Hadamard gap theorem condition has natural boundary?

EDIT: adding and multiplying a a function with domain containing the closed unit disk has to be regards as an equivalent example,that is example is the same one up to adding and multiplying a a function with domain containing the closed unit disk

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    $\begingroup$ One example is $$G(z)=\sum_{n=1}^{\infty}\frac{z^n}{1-z^n}$$ It may be written as $$G(z)=z + 2z^2 + 2z^3 + 3z^4 + 2z^5 + 4z^6 + \dots =\sum_{n=1}^{\infty}\tau (n)x^n$$$\tau (n)$is the number of divisors of $n$ ,The function has unit circle as a natural boundary, but does not satisfy Fabry or Hadamard gap theorem condition. $\endgroup$ Jul 28, 2014 at 6:44
  • $\begingroup$ Another is not in the class of what Fabry or Hadamard gap theorem concerns,but it is very similar to the example above. here,I quote from a master thesis:Poincare simple poles,Let $L$ be a smooth closed curve which bounds a convex set in the plane, $\{z_n\}$ be a sequence of dense distinct points on $L$, and $\{c_n\}$ be an absolutely summable sequence of non-zero complex numbers. define $f$ to be the function$$f(z)=\sum_{i=1}^{\infty}\frac{c_n}{z-z_n} \text{,where }z\notin L$$Then $f(z)$ ($f(z)$ restricted to the interior of $L$) does not have an analytic continuation across any point of $L$ $\endgroup$ Jul 28, 2014 at 6:51
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    $\begingroup$ This is extremely broad. Adding a function with domain containing the closed unit disk to a lacunary function doesn't change its natural boundary. To mention a huge family of very restricted, atypical lacunary functions, nonconstant functions on cusped finite volume Riemann surfaces lift to lacunary functions on the universal cover, and these generally don't satisfy the Fabry or Hadamard gap conditions. $\endgroup$ Jul 28, 2014 at 19:22
  • $\begingroup$ @DouglasZare,thanks,why don't you make your comment as an answer,and I have to edit my post in some words to say that adding and multiplying a a function with domain containing the closed unit disk has to be regards as an equivalent example,that is example is the same one up to adding and multiplying a a function with domain containing the closed unit disk $\endgroup$ Jul 29, 2014 at 2:00
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    $\begingroup$ Another class of function that don't satisfy the Fabry or Hadamard gap conditions: Theorem (Szeg$\ddot{o}$, 1922). If $$f (z) = \sum a_nz_n$$ and the set of values of ${a_n}$ is a finite set, then either $|z| = 1$ is a natural boundary, or else an is eventually periodic, in which case $f$is a rational function with poles on $\mathbb{\partial D}$. $\endgroup$ Jul 29, 2014 at 3:30

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