On properties of finite-order meromorphic solutions of a certain difference Painlevé I equation (Q2891975)

The difference Painlevé equation I NEWLINE\[NEWLINE f(z+1)+f(z)+f(z-1)=\frac{az+b}{f(z)}+c NEWLINE\]NEWLINE is considered, where \(a\), \(b\) and \(c\) are constants with \(|a|+|b|\neq0\). For a meromorphic function \(f\), the authors denote by \(\sigma(f)\) and \(\lambda(f)\) the growth order of \(f\) and the exponent of convergence of zeros of \(f\), respectively. One of the main results is the following theorem:NEWLINENEWLINE If the Painlevé equation I possesses a transcendental meromorphic solution \(f\) of finite order, then:NEWLINENEWLINE(i) \(\lambda(1/f)=\lambda(f)=\sigma(f)\);NEWLINENEWLINE(ii) \(\lambda(f-P)=\sigma(f)\) for any non constant polynomial \(P\);NEWLINENEWLINE(iii) \(f\) has no Borel exceptional value if \(a\neq0\); the Borel exceptional value of \(f\) can only come from a set \(\{z\;|\;3z^2-cz-b=0\}\) if \(a=0\).NEWLINENEWLINE One of the tools for the proof is the difference analogue of Clunie's lemma obtained in [\textit{I. Laine} and \textit{C. C. Yang}, ``Clunie theorems for difference and \(q\)-difference polynomials'', J. Lond. Math. Soc., II. Ser. 76, No. 3, 556--566 (2007; Zbl 1132.30013)]. The authors also consider rational solutions of the given equation.