Analytic solutions of an iterative functional differential equation

This paper is concerned with an iterative functional differential equation x′(z) = x(m)(z), where x(m)(z) = x(x(…x(z))) is the mth iterate of the function x(z). By constructing a convergent power series solution y(z) of a companion equation of the form αy′(αz) = y′(z)y(αmz), analytic solutions of the form y(αy−1(z)) for the original differential equation are obtained.