Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

Finite order solutions of difference equations, and difference Painlevé equations IV


Author: Zhi-Tao Wen
Journal: Proc. Amer. Math. Soc. 144 (2016), 4247-4260
MSC (2010): Primary 39A10; Secondary 30D35, 39A12
Published electronically: June 3, 2016
MathSciNet review: 3531176
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

In this paper, from the non-linear difference equation

$\displaystyle (\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

$\displaystyle (\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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 39A10, 30D35, 39A12

Retrieve articles in all journals with MSC (2010): 39A10, 30D35, 39A12


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
Email: zhitaowen@gmail.com

DOI: https://doi.org/10.1090/proc/13210
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
Article copyright: © Copyright 2016 American Mathematical Society