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Volume 48 Number 4 Volume 48 Number 5 Volume 48 Number 6

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Research articles

ScienceAsia 48 (2022): 681-686 |doi: 10.2306/scienceasia1513-1874.2022.091


Fermat type functional equations, several complex variables and Euler operator


Liu Yanga,*, Li-Xiong Shib, Sheng-Yao Zhoua

 
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, E-mail: yangliu6@ahut.edu.cn

Received 12 Dec 2020, Accepted 7 May 2022