Analytic solutions of a polynomial-like iterative functional equation

This paper is concerned with a polynomial-like iterative functional equation lambda1f(z) + lambda2f(2)(z) + .. . . + lambdamf[m](z) = F(z), where f(z) is an unknown function, F(z) is a given function, f[i]l denotes the i-th iterate of f, and lambda1, lambda2, . . .,lambdam are complex constants. By constructing a convergent power series solution phi(z) of an auxiliary equation of the form lambda1alpha(alphaz)+lambda2phi(alpha2z)+.....+lambdamphi(alphamz)=F(phi(z)), analytic solutions of the form phi(alphaphi-1(z)) for the original iterative functional equation are obtained.