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Existence of zero-order meromorphic solutions in detecting $ q$-difference Painlevé equations


Authors: Risto Korhonen and Zhi-Tao Wen
Journal: Trans. Amer. Math. Soc. 368 (2016), 4993-5008
MSC (2010): Primary 39A13; Secondary 33E17, 30D35
DOI: https://doi.org/10.1090/tran/6491
Published electronically: October 28, 2015
MathSciNet review: 3456168
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Abstract: The existence of zero-order meromorphic solutions is used as a sufficient condition in detecting $ q$-difference equations of Painlevé type. It is shown that demanding the existence of at least one non-rational zero-order meromorphic solution $ w(z)$ is sufficient to reduce a canonical class of $ q$-difference equations with rational coefficients into a short list of Painlevé type $ q$-difference equations, unless $ w(z)$ is simultaneously a solution of a $ q$-difference Riccati equation of a specific form.


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Additional Information

Risto Korhonen
Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
Email: risto.korhonen@uef.fi

Zhi-Tao Wen
Affiliation: Department of Mathematics, Taiyuan University of Technology, 030024, Taiyuan, People’s Republic of China
Email: zhitaowen@gmail.com

DOI: https://doi.org/10.1090/tran/6491
Keywords: Painlev\'e, $q$-difference equation, meromorphic, Nevanlinna theory
Received by editor(s): September 13, 2013
Received by editor(s) in revised form: June 3, 2014
Published electronically: October 28, 2015
Additional Notes: The first author was supported in part by the Academy of Finland grant number 268009, and the second author was supported in part by China Scholarship Council (CSC)
Article copyright: © Copyright 2015 American Mathematical Society