On properties of meromorphic solutions for some difference equations (Q638735)

The author studies value distribution of finite-order meromorphic solutions of the equation \[ w(z+1)+w(z-1) = \frac{az+b}{w(z)}+\frac{c}{w(z)^2}, \quad a,b,c\in\mathbb{C}, \] which is known as one of the alternate forms of the difference Painlevé I equation and of \[ w(z+1)+w(z-1) = \frac{(az+b)w(z)}{1-w(z)^2},\quad a,b\in\mathbb{C}, \] which is a special case of the difference Painlevé equation II. If \(a=0\), then these equations are known to be solvable in terms of elliptic functions, so in this case the existence of finite-order meromorphic solutions is clear. In the case \(a\not=0\), \textit{S. Shimomura} [J. Phys. A, Math. Theor. 42, No. 31, Article ID 315213, 19 p. (2009; Zbl 1188.39023)] has shown that meromorphic solutions exist, at least under certain restrictions to the coefficients, but the question of the existence of finite-order solutions is still open in the general case. In the paper under review the author finds, assuming that the solutions exist, all the possible Borel exceptional values of finite-order meromorphic solutions of the equations above. In addition, the author studies the properties of rational solutions of these equations. \textit{K. Kajiwara, K. Yamamoto} and \textit{Y. Ohta} [Phys. Lett., A 232, No. 3--4, 189--199 (1997; Zbl 1053.39500)] have also considered rational solutions of Painlevé equations in the discrete case. They have constructed the rational solutions of the discrete Painlevé~II equation based on bilinear formalism.