Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2015; 30(4): 447-456

Printed October 31, 2015

https://doi.org/10.4134/CKMS.2015.30.4.447

Copyright © The Korean Mathematical Society.

Entire solutions of differential-difference equation and Fermat type $q$-difference differential equations

Min Feng Chen and Zong Sheng Gao

Beihang University, Beihang University

Abstract

In this paper, we investigate the differential-difference equation $$(f(z+c)-f(z))^{2}+P(z)^{2}(f^{(k)}(z))^{2}=Q(z),$$ where $P(z),~Q(z)$ are nonzero polynomials. In addition, we also investigate Fermat type $q$-difference differential equations $$f(qz)^{2}+(f^{(k)}(z))^{2}=1\quad \text{and} \quad (f(qz)-f(z))^{2}+(f^{(k)}(z))^{2}=1.$$ If the above equations admit a transcendental entire solution of finite order, then we can obtain the precise expression of the solution.

Keywords: differential-difference equation, Fermat type $q$-difference differential equations, transcendental entire solution, finite order

MSC numbers: 39B32, 34M05, 30D35