Abstract
In this paper, we investigate the following two Painlevé III equations: \(\overline{w}\underline{w}(w^{2}-1)=w^{2}+\mu\) and \(\overline{w}\underline{w}(w^{2}-1)=w^{2}-\lambda w\), where \(\overline{w}:=w(z+1)\), \(\underline{w}:=w(z-1)\) and \(\mu\) (\(\mu\neq-1\)) and \(\lambda\notin\{\pm 1\}\) are constants. We discuss the questions of existence of rational solutions, of Borel exceptional values and the exponents of convergence of zeros, poles and fixed points of transcendental meromorphic solutions of these equations.
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ACKNOWLEDGMENTS
The authors would like to thank an Associate Editor and a referee for their valuable suggestions that led to considerable improvement of the paper.
Funding
This research was supported by the NNSF of China, grants nos. 11201014, 11171013, 11126036, and by the Fundamental Research for the Central University.
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Liu, L., Zhang, J. Some Results on the Painlevé III Difference Equations with Constant Coefficients. J. Contemp. Mathemat. Anal. 55, 115–125 (2020). https://doi.org/10.3103/S1068362320020065
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DOI: https://doi.org/10.3103/S1068362320020065