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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.
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참고문헌 (15)
참고문헌 신청
C. C. Yang / 1970 / A generalization of a theorem of P. Montel on entire functions / Proc. Amer. Math. Sci 26 : 332 ~ 334
C. C. Yang / 2003 / Mathematics and its Applications / Kluwer Academic Publishers Group
C. C. Yang / 2010 / On analogies between nonlinear difference and differential equations / Proc. Japan. Acad. Ser. A. Math. Sci 86 : 10 ~ 14
D. C. Barnett / 2007 / Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations / Proc. Roy. Soc. Edinburgh Sect. A 137 (3) : 457 ~ 474
F. Gross / 1966 / On the equation f(z)n + g(z)n = h(z)n / Amer. Math. Monthly 73 : 1093 ~ 1096
C. C. Yang / 2003 / Mathematics and its Applications / Kluwer Academic Publishers Group
C. C. Yang / 2010 / On analogies between nonlinear difference and differential equations / Proc. Japan. Acad. Ser. A. Math. Sci 86 : 10 ~ 14
D. C. Barnett / 2007 / Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations / Proc. Roy. Soc. Edinburgh Sect. A 137 (3) : 457 ~ 474
F. Gross / 1966 / On the equation f(z)n + g(z)n = h(z)n / Amer. Math. Monthly 73 : 1093 ~ 1096
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