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Growth and zeros of meromorphic solution of some linear difference equations. (English) Zbl 1208.39028

The article deals with the growth of meromorphic solutions and the growth of their zeros for linear difference equation

${P}_{n}\left(z\right)f\left(z+n\right)+\cdots +{P}_{1}\left(z\right)f\left(z+1\right)+{P}_{0}\left(z\right)f\left(z\right)=F\left(z\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $F\left(z\right),{P}_{0}\left(z\right),{P}_{1}\left(z\right),\cdots ,{P}_{n}\left(z\right)$ are polynomials with $F{P}_{0}{P}_{n}¬\equiv 0$ and

$deg\left({P}_{n}+\cdots +{P}_{1}+{P}_{0}\right)=max\phantom{\rule{4pt}{0ex}}\left\{deg{P}_{j}:\phantom{\rule{4pt}{0ex}}j=0,1,\cdots ,n\right\}\ge 1·\phantom{\rule{2.em}{0ex}}\left(2\right)$

The following three theorems are proved: (i) If (2) holds then every finite order transcendental meromorphic solution $f\left(z\right)$ to (1) satisfies $\sigma \left(f\right)\ge 1$ and $\sigma \left(f\right)=\lambda \left(f\right)$; (ii) If (2) holds and $F\left(z\right)\equiv 0$ then every finite order transcendental meromorphic solution $f\left(z\right)$ to (1) satisfies $\sigma \left(f\right)\ge 1$, takes every non-zero value $a\in ℂ$ infinitely often and $\sigma \left(f\right)=\lambda \left(f-a\right)$; (iii) Without assumption (2) every meromorphic solution $f\left(z\right)$ to (1) with infinitely many poles satisfies $\sigma \left(f\right)\ge 1$. In these results $\sigma \left(f\right)$ is the order of growth of meromorphic function $f\left(z\right)$ and $\lambda \left(f\right)$ the exponent of the growth of zeros of $f\left(z\right)$. In the article there are some simple illustrative examples.

##### MSC:
 39A45 Difference equations in the complex domain 39A06 Linear equations (difference equations) 30D05 Functional equations in the complex domain, iteration and composition of analytic functions
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