zbMATH — the first resource for mathematics

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)].

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