Hexagonal circle patterns with constant intersection angles and discrete Painleve and Riccati equations

Hexagonal circle patterns with constant intersection angles mimicking holomorphic maps z^c and log(z) are studied. It is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete Painleve and Riccati equations. The general solution of the Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete z^c and log(z), are established.