Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE EXISTENCE OF SOLUTIONS OF FERMAT-TYPE DIFFERENTIAL-DIFFERENCE EQUATIONS

  • Chen, Jun-Fan (School of Mathematics and Statistics & Fujian Key Laboratory of Mathematical Analysis and Applications Fujian Normal University) ;
  • Lin, Shu-Qing (School of Mathematics and Statistics & Fujian Key Laboratory of Mathematical Analysis and Applications Fujian Normal University)
  • Received : 2020.09.09
  • Accepted : 2021.03.08
  • Published : 2021.07.31
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Abstract

We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations [f(z)f'(z)]n + P2(z)fm(z + 𝜂) = Q(z) and [f(z)f'(z)]n + P(z)[∆𝜂f(z)]m = Q(z), where P(z) and Q(z) are non-zero polynomials, m and n are positive integers, and 𝜂 ∈ ℂ \ {0}. In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation P2(z) [f(k)(z)]2 + [αf(z + 𝜂) - 𝛽f(z)]2 = er(z), where $P(z){\not\equiv}0$ is a polynomial, r(z) is a non-constant polynomial, α ≠ 0 and 𝛽 are constants, k is a positive integer, and 𝜂 ∈ ℂ \ {0}. Our results generalize some previous results.

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Acknowledgement

The authors would like to thank the referees for their thorough comments and helpful suggestions.

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