# Deficiency Values for the Solutions of Differential Equations with Branching Point

### Abstract

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.

Published

25.07.2014

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 66, no. 7, July 2014, pp. 939–957, http://umj.imath.kiev.ua/index.php/umj/article/view/2190.

Issue

Section

Research articles