Transcendental Solutions of a Class of Minimal Functional Equations

http://dx.doi.org/10.4153/CMB-2011-157-x
Canad. Math. Bull. 56(2013), 283-291

  Published:2011-08-03
 Printed: Jun 2013

We prove a result concerning power series $f(z)\in\mathbb{C}[\mkern-3mu[z]\mkern-3mu]$ satisfying a functional equation of the form $$ f(z^d)=\sum_{k=1}^n \frac{A_k(z)}{B_k(z)}f(z)^k, $$ where $A_k(z),B_k(z)\in \mathbb{C}[z]$. In particular, we show that if $f(z)$ satisfies a minimal functional equation of the above form with $n\geqslant 2$, then $f(z)$ is necessarily transcendental. Towards a more complete classification, the case $n=1$ is also considered.