Transcendental Solutions of a Class of Minimal Functional Equations

Coons, Michael

Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 283-291

Résumé

We prove a result concerning power series $f\left( Z \right)\,\in \,\mathbb{C}\left[\!\left[ Z \right]\!\right]$ satisfying a functional equation ofthe form? 1 $$f\left( {{Z}^{d}} \right)\,=\,\sum\limits_{k=1}^{n}{\frac{{{A}_{k}}\left( Z \right)}{{{B}_{k}}\left( Z \right)}\,f{{\left( Z \right)}^{k}}},$$ ,where ${{A}_{k}}\left( Z \right),\,{{B}_{k}}\left( Z \right)\,\in \,\mathbb{C}\left[ Z \right]$ . In particular, we show that if $f\left( Z \right)$ satisfies a minimal functional equation of the above form with $n\,\ge \,2$ , then $f\left( Z \right)$ is necessarily transcendental. Towards a more complete classification, the case $n=\,1$ is also considered.

Détail

DOI : 10.4153/CMB-2011-157-x

Mots-clés : 11B37, 11B83, 11J91, transcendence, generating functions, Mahler-type functional equation

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