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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

$\sum _{j=1}^{n}{d}_{j}\left(z\right)y\circ {p}_{j}=\frac{{a}_{0}\left(z\right)+{a}_{1}\left(z\right)y\circ {p}_{n+1}+\cdots +{a}_{t}\left(z\right){\left(y\circ {p}_{n+1}\right)}^{t}}{{b}_{0}\left(z\right)+{b}_{1}\left(z\right)y\circ {p}_{n+1}+\cdots +{b}_{s}\left(z\right){\left(y\circ {p}_{n+1}\right)}^{s}},\phantom{\rule{2.em}{0ex}}\left(1\right)$

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

In case ${p}_{j}\left(z\right)={q}^{j}z$, $j=1,2,\cdots ,N+1$, 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:
 30D35 Distribution of values (one complex variable); Nevanlinna theory 39A13 Difference 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 [3] Eremenko, A. E.; Sodin, M. L.: Iterations of rational functions and the distribution of the values of Poincaré function, Teor. funktsiĭ funktsional. Anal. i prilozhen. 53, 18-25 (1990) · Zbl 0735.30029 [4] Goldstein, R.: Some results on factorization of meromorphic functions, J. lond. Math. soc. (2) 4, 357-364 (1971) · Zbl 0223.30036 · doi:10.1112/jlms/s2-4.2.357 [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) [13] Valiron, G.: Foncions analytiques, (1952) [14] Yang, C. -C.; Yi, H. -X.: Uniqueness theory of meromorphic functions, (2003) [15] Zhang, J.; Korhonen, R.: On the Nevanlinna characteristic of $f\left(qz\right)$ 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