We study the distribution of values of the solutions of an algebraic differential equation P(z, f, f′, . . . , f (s)) = 0 with the property that its coefficients and solutions have a branching point at infinity (e.g., a logarithmic singularity). It is proved that if a ∈ ℂ is a deficiency value of f and f grows faster than the coefficients, then the following identity takes place: P(z, a, 0, . . . , 0) ≡ 0, z ∈ {z : r 0 ≤ |z| < ∞}. If P(z, a, 0, . . . , 0) is not identically equal to zero in the collection of variables z and a, then only finitely many values of a can be deficiency values for the solutions f ∈ M b with finite order of growth.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 7, pp. 939–957, July, 2014.
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Mokhon’ko, A.A., Mokhon’ko, A.Z. Deficiency Values for the Solutions of Differential Equations with Branching Point. Ukr Math J 66, 1048–1069 (2014). https://doi.org/10.1007/s11253-014-0994-8
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DOI: https://doi.org/10.1007/s11253-014-0994-8