Research articles
ScienceAsia 48 (2022): 681686 doi:
10.2306/scienceasia15131874.2022.091
Fermat type functional equations, several complex variables
and Euler operator
Liu Yang^{a,*}, LiXiong Shi^{b}, ShengYao Zhou^{a}
ABSTRACT: We describe the entire solutions for Fermat type functional equations with functional coefficients in C
n
,
i.e., hf p+kgq = 1, where p, q ? 2 are two integers. We then apply the result to obtain that entire function solutions f , g
of f
2 + g
2 = 1 in C
n
are constant if D f ?1
(0) ? Dg?1
(0) with ignoring multiplicities, where D :=
Pn
j=1
zj
?
? zj
is the Euler
operator. Meromorphic function solutions of f
3 + g
3 = 1 in C
n
and applications to nonlinear (ordinary and partial)
differential equations are also discussed.
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^{a} 
School of Mathematics and Physics, Anhui University of Technology, Maanshan 243032 China 
^{b} 
College of Science, Yunnan Agricultural University, Kunming 650201 China 
* Corresponding author, Email: yangliu6@ahut.edu.cn
Received 12 Dec 2020, Accepted 7 May 2022
