J. Aust. Math. Soc. 83 (2007), no. 3, pp. 369–384.

# Permutable functions concerning differential equations

 X. Hua† R. Vaillancourt X. L. Wang Department of Mathematics and StatisticsUniversity of Ottawa Ottawa, ON, K1N 6N5 Canadahua@mathstat.uottawa.ca Department of Mathematics and StatisticsUniversity of Ottawa Ottawa, ON, K1N 6N5Canadaremi@ottawa.ca Department of Applied MathematicsNanjing University of Finance and EconomicsNanjing 210003, JiangsuChinawangxiaoling@vip.163.com
Received 25 January 2006; revised 28 June 2006
Communicated by P. Fenton
This research was partially supported by the Natural Sciences and Engineering Research Council of Canada
and NNSF of China, No: 10371069.

## Abstract

Let f and g be two permutable transcendental entire functions. Assume that f is a solution of a linear differential equation with polynomial coefficients. We prove that, under some restrictions on the coefficients and the growth of f and g, there exist two non-constant rational functions R_1 and R_2 such that R_1(f)=R_2(g). As a corollary, we show that f and g have the same Julia set: J(f)=J(g). As an application, we study a function f which is a combination of exponential functions with polynomial coefficients. This research addresses an open question due to Baker.

 2000 Mathematics Subject Classification: primary 30D05, 37F10, 37F50; secondary 34A20 (Metadata: XML, RSS, BibTeX) †indicates author for correspondence

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