On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point

Abstract

Assume that the coefficients and solutions of the equation f (n)+p n−1(z)f (n−1)+. . .+p s+1(z)f (s+1)+ . . . + p 0(z)f = 0 have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients p j , j = s+1, . . . ,n−1, increase slower (in terms of the Nevanlinna characteristics) than p s (z). It is proved that this equation has at most s linearly independent solutions of finite order.