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 Statistics
University of Ottawa
Ottawa, ON, K1N 6N5
Canada
hua@mathstat.uottawa.ca		 Department of Mathematics and Statistics
University of Ottawa
Ottawa, ON, K1N 6N5
Canada
remi@ottawa.ca		 Department of Applied Mathematics
Nanjing University of Finance and Economics
Nanjing 210003, Jiangsu
China
wangxiaoling@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.

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2000 Mathematics Subject Classification: primary 30D05, 37F10, 37F50; secondary 34A20
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†indicates author for correspondence

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