Finite order solutions of difference equations, and difference Painlevé equations IV
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Abstract:
In this paper, from the non-linear difference equation \[ (\overline {w}+w)(w+\underline {w})=\frac {P(z,w)}{Q(z,w)} \] where $P(z,w)$ and $Q(r,w)$ are polynomials in $w(z)$ without common factors having small function coefficients related to $w(z)$, we present the form of difference Painlevé equation IV \[ (\overline {w}+w)(w+\underline {w})=\frac {(w^2-a^2)(w^2-b^2)}{(w+(\alpha z+\beta ))^2+\pi }, \] where $a$, $b$, $\alpha$, $\beta$ and $\pi$ are period small functions related to $w$. It shows that if the above difference equation admits at least one meromorphic solution $w(z)$ of finite order, then the difference equation can be transformed by Möbius tranformation in $w$ to difference Painlevé IV, unless $w$ is the solution of difference Riccati equations.References
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Additional Information
- Zhi-Tao Wen
- Affiliation: Department of Mathematics, Taiyuan University of Technology, No. 79 Yingze West Street, 030024 Taiyuan, People’s Republic of China
- MR Author ID: 917720
- Email: zhitaowen@gmail.com
- Received by editor(s): August 30, 2014
- Received by editor(s) in revised form: September 25, 2015
- Published electronically: June 3, 2016
- Additional Notes: This project was supported by the National Natural Science Foundation for the Youth of China (No. 11501402) and Shanxi Scholarship Council of China (No. 2015-043)
- Communicated by: Yingfei Yi
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4247-4260
- MSC (2010): Primary 39A10; Secondary 30D35, 39A12
- DOI: https://doi.org/10.1090/proc/13210
- MathSciNet review: 3531176