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Hexagonal circle patterns with constant intersection angles mimickingholomorphic maps z^c and log(z) are studied. It is shown that the correspondingcircle patterns are immersed and described by special separatrix solutions ofdiscrete Painleve and Riccati equations. The general solution of the Riccatiequation is expressed in terms of the hypergeometric function. Globalproperties of these solutions, as well as of the discrete z^c and log(z), areestablished.Comment: 18 pages, 6 figure

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