We study the differential equations w2＋ R（z）（w（k））2 = Q（z）, where R（z）, Q（z） are nonzero rationalfunctions. We prove （1） if the differential equation w2 ＋R（z）（w＇）2= Q（z）, where R（z）, Q（z） ar...We study the differential equations w2＋ R（z）（w（k））2 = Q（z）, where R（z）, Q（z） are nonzero rationalfunctions. We prove （1） if the differential equation w2 ＋R（z）（w＇）2= Q（z）, where R（z）, Q（z） are nonzero rational functions, admits a transcendental meromorphic solution f, then Q= C （constant）, the multiplicities of the zeros of R（z） are no greater than 2 and f（z） = √C cos α（z）, where α（z） is a primitive of 1 such that √C cos α（z） is a transcendental meromorphic function. （2） if the differential equation w2 ＋ R（z）（w（k））2 = Q（z）, where k ≥ 2 is an integer and R, Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C （constant）, It（z） ≡ A （constant） and f（z） = √C cos（az ＋ b）, where a2k = A1/A.展开更多
基金Supported by the National Natural Science Science Foundation of China（10671056） Acknowledgements couragement and numerous this paper. The authors are grateful to Professor Xu Guang-shan for his envaluable suggestions on the improvement of the original draft of
文摘We prove the transcendence for the values of Mahler type function with several variables, which satisfies some non-linear functional equation.
基金supported by National Natural Science Foundation of China(Grant Nos.10871011 and 11271179)
文摘We study the differential equations w2＋ R（z）（w（k））2 = Q（z）, where R（z）, Q（z） are nonzero rationalfunctions. We prove （1） if the differential equation w2 ＋R（z）（w＇）2= Q（z）, where R（z）, Q（z） are nonzero rational functions, admits a transcendental meromorphic solution f, then Q= C （constant）, the multiplicities of the zeros of R（z） are no greater than 2 and f（z） = √C cos α（z）, where α（z） is a primitive of 1 such that √C cos α（z） is a transcendental meromorphic function. （2） if the differential equation w2 ＋ R（z）（w（k））2 = Q（z）, where k ≥ 2 is an integer and R, Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C （constant）, It（z） ≡ A （constant） and f（z） = √C cos（az ＋ b）, where a2k = A1/A.
