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Transcendental Solutions of a Class of Minimal Functional Equations

Published online by Cambridge University Press:  20 November 2018

Michael Coons
Michael Coons*
Affiliation:
University of Waterloo, Dept. of Pure Mathematics, Waterloo, ON, N2L 3G1 e-mail: mcoons@math.uwaterloo.ca
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Abstract

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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 of the 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.

Keywords

11B3711B8311J91transcendencegenerating functionsMahler-type functional equation
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 283 - 291
Copyright
Copyright © Canadian Mathematical Society 2013

References

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