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COMPLETE CLASSIFICATION OF MEROMORPHIC SOLUTIONS OF A FERMAT-TYPE DIFFERENCE EQUATION
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 30 October 2026, pp. 1-9
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We investigate meromorphic solutions of the difference equation
$$ \begin{align*} f(z)^2+f(z+c)=h(z), \quad c\in\mathbb C\setminus\{0\}, \end{align*} $$where h is a transcendental meromorphic function satisfying $ N(r,h)+N(r,{1}/{h})=S(r,h). $
We prove that every solution of hyper-order less than $1$
is of the form $$ \begin{align*} f(z)=e^{az+b}-1, \quad h(z)=e^{2az+2b}, \end{align*} $$where $a,b\in \mathbb C$
and $e^{ac}=2$
. This gives a negative answer to the open question posed by Laine and Latreuch [‘Notes on Fermat-type difference equations’, Bull. Aust. Math. Soc. 111(2) (2025), 324–335] concerning the possible realisation of the second alternative in their Theorem 2.2 (see Remark 2.3(c)). We also construct a solution of hyper-order $1$
, showing that this restriction is sharp.
