## Differential polynomials generated by linear differential equations.(English)Zbl 1080.34076

The authors study the value distribution theory of differential polynomials generated by solutions of linear differential equations. Let $${\mathcal L}$$ be a differential subfield of the field of meromorphic functions in a domain $$G\subset{\mathbb C}$$. For a polynomial $$P\in{\mathcal L}[y_0,y_1,\dots,y_r]$$, they consider the differential polynomial $$P[f]=P(f,f',\dots,f^{(r)})$$ given by a solution $$f(z)$$ of the differential equation $f''+A(z)f=0.$ Here, $$A(z)$$ is assumed to be transcendental meromorphic, transcendental entire or a polynomial. Under some conditions for value distribution natures of $${\mathcal L}, A, f$$, they evaluate the iterated exponents of convergence for the fixed points of $$P[f]$$. The results can be regarded as extensions of the results due to S. B. Bank [Proc. Lond. Math. Soc., III. Ser. 50, 505–534 (1985; Zbl 0545.30022)] and J. Wang and H.-X. Yi [Complex Variables, Theory Appl. 48, 83–94 (2003; Zbl 1071.30029)].

### MSC:

 34M99 Ordinary differential equations in the complex domain 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

### Keywords:

Nevanlinna theory; iterated order of growth

### Citations:

Zbl 0545.30022; Zbl 1071.30029
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