Solutions of Fermat-Type Partial Differential-Difference Equations in Cn

For two meromorphic functions f and g, the equation f(m)+g(m)=1 can be regarded as Fermat-type equations. Using Nevanlinna theory for meromorphic functions in several complex variables, the main purpose of this paper is to investigate the properties of the transcendental entire solutions of Fermat-type difference and partial differential-difference equations in C-n. In addition, we find the precise form of the transcendental entire solutions in C-2 with finite order of the Fermat-type partial differential-difference equation(& part;f(z(1),z(2))/& part;z(1))(2)+(f(z(1)+c1,z(2)+c(2))-f(z(1),z(2)))(2)=1andf(2)(z(1),z(2))+P-2(z(1),z(2)(& part;f(z(1)+c1,z(2)+c(2))& part;z(1)-& part;f(z(1),z(2))& part;z(1))(2)=1,where P(z(1),z(2)) is a polynomial in C2. Moreover, one of the main results of the paper significantly improved the result of Xu and Cao [Mediterr. J. Math. (2018) 15:227 , 1-14 and Mediterr. J. Math. (2020) 17:8, 1-4].

Automated summary

This paper investigates the properties of transcendental entire solutions of Fermat-type difference and partial differential-difference equations in C-n using Nevanlinna theory for meromorphic functions in several complex variables. It also provides the precise form of transcendental entire solutions in C-2 with finite order of the Fermat-type partial differential-difference equation and improves upon previous results.