Discrete functional equations in complex domains

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By means of the value distribution theory, we investigated ordinary differential equations and discrete functional equations in complex domains, in particular, linear difference equations having polynomial coefficients and the Schroeder functional equations. We obtain results on Malmquist-Yoshida type theorems for nonlinear differential equations in connection with complex dynamics theory, and results on linear differential equation of the second order with elliptic coefficients. We also constructed Wiman-Valiron theory for binomial series in order to treat linear difference equations. The Schroeder equations are considered from the point of view of the connections between the Nevanlinna theory and the complex dynamics theory. In particular, we gave a proof of the result on Borel directions of the Schroeder functions and the Julia set of the rational functions from which the Schroeder functions are generated. Below we report the researches by each investigator.Mori obtained uniqueness … More theorems on some complex domains which are bounded or countable.Shimomura investigated Painleve transcendents. He obtained the lower estimate on meromorphic solutions of higher order Painleve equations. Modified type of Painleve equations (III), and (V) are considered. He also gave nontrivial examples on nonlinear differential equations having the Painleve properties.Morosawa considered the parameter space of complex dynamics on the complex error functions. He obtained the topological properties of the Julia set of the hyperbolic complex error function.Fermat type functional equations are treated by Tohge. With Professor Gundersen (New Orleans), he found a new entire function satisfying a Fermat type equation. He also investigated the value distribution theory in angular domains.Sawada obtained an algebroid function with three sheets on which analytic functions have deficiencies restricted by some given conditions. He also treated meromorphic solutions of some algebraic differential equations. Less