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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
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