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