Meromorphic continuation of functions given by limit k-periodic continued fractions

A continued fraction whose elements depend on a variable z defines a function for all the z which makes the continued fraction converge. If the convergence is locally uniform (with respect to the chordal metric) in some domain, then the function is meromorphic (or constant) in this domain. By using modified approximants, one can, under appropriate conditions, extend the domain where the continued fraction converges locally uniformly.