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DISCRETE RICCATI EQUATION, HYPERGEOMETRIC FUNCTIONS AND CIRCLE PATTERNS OF SCHRAMM TYPE
Published online by Cambridge University Press: 14 July 2005
Abstract
Square grid circle patterns with prescribed intersection angles, mimicking holomorphic maps $z^{\gamma }$ and ${\rm log}(z)$ are studied. It is shown that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlevé and Riccati equations. The 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 $z^{\gamma }$ and ${\rm log}(z)$, are established.
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- Research Article
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- 2005 Glasgow Mathematical Journal Trust
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