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Transcendental Solutions of a Class of Minimal Functional Equations

  • Michael Coons (a1)

Abstract

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.

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References

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Transcendental Solutions of a Class of Minimal Functional Equations

  • Michael Coons (a1)

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