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# Transcendental Solutions of a Class of Minimal Functional Equations

Published:2011-08-03
Printed: Jun 2013
• Michael Coons,
University of Waterloo, Dept. of Pure Mathematics, Waterloo, ON, N2L 3G1
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## Abstract

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.
 Keywords: transcendence, generating functions, Mahler-type functional equation
 MSC Classifications: 11B37 - Recurrences {For applications to special functions, see 33-XX} 11B83 - Special sequences and polynomials 11J91 - Transcendence theory of other special functions