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Growth and poles of meromorphic solutions of some difference equations. (English) Zbl 1222.30023

By means of the Nevanlinna theory, the author treats functional equations of the form

j=1 n d j (z)yp j =a 0 (z)+a 1 (z)yp n+1 ++a t (z)(yp n+1 ) t b 0 (z)+b 1 (z)yp n+1 ++b s (z)(yp n+1 ) s ,(1)

where p j , j=1,2,,n+1, are polynomials of degree k j >0 with p j 0. The author studies growth properties and the pole distribution of meromorphic solutions of (1).

In case p j (z)=q j z, j=1,2,,N+1, the functional equation is known as a q-difference equation. Recent developments in the study of nonlinear q-difference equations in the complex plane can be found in, e.g., [J. Zhang and R. Korhonen, J. Math. Anal. Appl. 369, No. 2, 537–544 (2010; Zbl 1198.30033)] and [X.-M. Zheng and Z.-X. Chen, ibid. 361, No. 2, 472–480 (2010; Zbl 1185.39006)]. The author obtaines generalizations of the results in the cited papers.

MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory
39A13Difference equations, scaling (q-differences)
References:
[1]Barnett, D.; Halburd, R. G.; Korhonen, R. J.; Morgan, W.: Application of Nevanlinna theory to q-difference equations, Proc. roy. Soc. Edinburgh sect. A 137, 457-474 (2007)
[2]Bergweiler, W.; Ishizaki, K.; Yanagihara, N.: Meromorphic solutions of some functional equations, Methods appl. Anal. 5, No. 3, 248-259 (1998) · Zbl 0924.39017
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[5]Goldstein, R.: On meromorphic solutions of certain functional equations, Aequationes math. 18, 112-157 (1978) · Zbl 0384.30009 · doi:10.1007/BF01844071
[6]Gundersen, G.: Finite order solutions of second order linear differential equations, Trans. amer. Math. soc. 305, 415-429 (1988) · Zbl 0669.34010 · doi:10.2307/2001061
[7]Gundersen, G.; Heittokangas, J.; Laine, I.; Rieppo, J.; Yang, D.: Meromorphic solutions of generalized Schröder equation, Aequationes math. 63, 110-135 (2002) · Zbl 1012.30016 · doi:10.1007/s00010-002-8010-z
[8]Hayman, W.: Meromorphic functions, (1964) · Zbl 0115.06203
[9]Hayman, W.: Angular value distribution of power series with gaps, Proc. lond. Math. soc. 24, 590-624 (1972) · Zbl 0239.30035 · doi:10.1112/plms/s3-24.4.590
[10]Ishizaki, K.; Yanagihara, N.: Deficiency for meromorphic solutions of Schröder functions, Complex var. Theory appl. 49, 539-548 (2004) · Zbl 1099.30012
[11]Ishizaki, K.; Yanagihara, N.: Borel and Julia direction of meromorphic Schröder functions, Math. proc. Cambridge philos. Soc. 139, 139-147 (2005) · Zbl 1076.30034 · doi:10.1017/S0305004105008492
[12]Laine, I.: Nevanlinna theory and complex differential equations, (1993)
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[14]Yang, C. -C.; Yi, H. -X.: Uniqueness theory of meromorphic functions, (2003)
[15]Zhang, J.; Korhonen, R.: On the Nevanlinna characteristic of f(qz) and its applications, J. math. Anal. appl. 369, 537-544 (2010) · Zbl 1198.30033 · doi:10.1016/j.jmaa.2010.03.038
[16]Zheng, X. -M.; Chen, Z. -X.: Some properties of meromorphic solutions of q-difference equations, J. math. Anal. appl. 361, 472-480 (2010) · Zbl 1185.39006 · doi:10.1016/j.jmaa.2009.07.009