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

Abstract

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.

18 pages, 6 figures

References in corpus (2)

• Hexagonal circle patterns and integrable systems: Patterns with the multi-ratio property and Lax equations on the regular triangular lattice
• Discrete Riccati equation, hypergeometric functions and circle patterns of Schramm type

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• Geometric description of discrete power function associated with the sixth Painlevé equation
• Hypergeometric $\tau$ Functions of the $q$-Painlevé Systems of Types $A_4^{(1)}$ and $(A_1+A_1^{\prime }{)}^{(1)}$