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