Discrete Riccati equation, hypergeometric functions and circle patterns of Schramm type

Square grid circle patterns with prescribed intersection angles, mimicking holomorphic maps z^a and log(z) are studied. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painleve and Riccati equations. General solution of this Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete z^a and log(z), are established.