Abstract
In this paper, we discuss the form and related properties of meromorphic solutions of hyper-order strictly less than 1 to Fermat type difference equation, and extend the previous results.
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Baker, I.N.: On a class of meromorphic functions. Proc. Am. Math. Soc. 17, 819–822 (1966)
Cartan, H.: Sur les záros des combinaisons lineaires de p fonctions holomorphes données. Math. Cluj 7, 5–31 (1933)
Chen, W., Han, Q., Liu, J.B.: On Fermat Diophantine functional equations, little Picard theorem and beyond. Aequ. Math. 93, 425–432 (2019)
Gross, F.: On the equation \(f^{n}+g^{n}=1\). Bull. Am. Math. Soc. 72, 86–88 (1966)
Gross, F.: On the equation \(f^{n}+g^{n}=h^n\). Am. Math. Mon. 73, 1093–1096 (1966)
Halburd, R.G., Korhonen, R.J., Tohge, K.: Holomorphic curves with shift-invariant hyperplane preimages. Trans. Am. Math. Soc. 366, 4267–4298 (2014)
Han, Q., Lü, F.: On the equation \(f^n(z) + g^n(z) = e^{\alpha z+\beta }\). J. Contemp. Math. Anal. 54, 98–102 (2019)
Iyer, G.: On certain functional equations. J. Indian Math. Soc. 3, 312–315 (1939)
Laine, I., Yang, C.C.: Clunie theorems for difference and q-difference polynomials. J. Lond. Math. Soc. 76, 556–566 (2007)
Liu, K.: Meromorphic functions sharing a set with applications to difference equations. J. Math. Anal. Appl. 359, 384–393 (2009)
Liu, K.: Existence of entire solutions of nonlinear difference equations. Czech. Math. J. 61, 565–576 (2011)
Liu, H.F., Mao, Z.Q.: Meromorphic solutions of certain nonlinear difference equations. Results Math. 76, 102 (2021)
Liu, K., Yang, L.Z.: On entire solutions of some differential-difference equations. Comput. Methods Funct. Theory 13, 433–447 (2013)
Liu, K., Cao, T.B., Cao, H.Z.: Entire solutions of Fermat type differential-difference equations. Arch. Math. 99, 147–155 (2012)
Lü, F., Guo, H.X.: On meromorphic solutions of the Fermat-Type functional equation \(f^n(z) + g^n(z) = e^{\alpha z+\beta }\). Mediterr. J. Math. 19, 118 (2022)
Lü, F., Han, Q.: On the Fermat-type equation \(f^3(z)+f^3(z +c) = 1\). Aequ. Math. 91, 129–136 (2017)
Mao, Z.Q., Liu, H.F.: On meromorphic solutions of nonlinear delay-differential equations. J. Math. Anal. Appl. 509, 125886 (2022)
Montel, P.: Lecons sur les Familles Normales de Fonctions Analytiques et Leurs Applications, pp. 135–136. Gauthier-Villars, Paris (1927)
Yanagihara, N.: Polynomial difference equations which have meromorphic solutions of finite order, analytic function theory of one complex variable. Pitman Res. Notes Math. Ser. 212, 368–392 (1989)
Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers, Dordrecht (2003)
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The work was supported by the NNSF of China (no. 12061042) and the NSF of Shandong Province (ZR2022MA071).
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XQ and LY drafted the manuscript, read and approved the final manuscript.
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Qi, X., Yang, L. On Meromorphic Solutions of the Fermat Type Difference Equations. Mediterr. J. Math. 21, 122 (2024). https://doi.org/10.1007/s00009-024-02643-y
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DOI: https://doi.org/10.1007/s00009-024-02643-y