Transcendental Solutions of a Class of Minimal Functional Equations
Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 283-291
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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.
Mots-clés :
11B37, 11B83, 11J91, transcendence, generating functions, Mahler-type functional equation
Coons, Michael. Transcendental Solutions of a Class of Minimal Functional Equations. Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 283-291. doi: 10.4153/CMB-2011-157-x
@article{10_4153_CMB_2011_157_x,
author = {Coons, Michael},
title = {Transcendental {Solutions} of a {Class} of {Minimal} {Functional} {Equations}},
journal = {Canadian mathematical bulletin},
pages = {283--291},
year = {2013},
volume = {56},
number = {2},
doi = {10.4153/CMB-2011-157-x},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-157-x/}
}
TY - JOUR AU - Coons, Michael TI - Transcendental Solutions of a Class of Minimal Functional Equations JO - Canadian mathematical bulletin PY - 2013 SP - 283 EP - 291 VL - 56 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-157-x/ DO - 10.4153/CMB-2011-157-x ID - 10_4153_CMB_2011_157_x ER -
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