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NOTES ON FERMAT-TYPE DIFFERENCE EQUATIONS

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  03 June 2024

ILPO LAINE Open the ORCID record for ILPO LAINE [Opens in a new window]  and
ZINELAABIDINE LATREUCH Open the ORCID record for ZINELAABIDINE LATREUCH [Opens in a new window]
ILPO LAINE
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101, Joensuu, Finland e-mail: [email protected]
ZINELAABIDINE LATREUCH*
Affiliation:
National Higher School of Mathematics, Scientific and Technology Hub of Sidi Abdellah, P.O. Box 75, Algiers 16093, Algeria
*
e-mail: [email protected]
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Abstract

We consider the existence problem of meromorphic solutions of the Fermat-type difference equation

$$ \begin{align*} f(z)^p+f(z+c)^q=h(z), \end{align*} $$

where $p,q$ are positive integers, and h has few zeros and poles in the sense that $N(r,h) + N(r,1/h) = S(r,h)$. As a particular case, we consider $h=e^g$, where g is an entire function. Additionally, we briefly discuss the case where h is small with respect to f in the standard sense $T(r,h)=S(r,f)$.

Keywords

meromorphic functionsNevanlinna theoryFermat-type difference equations

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 39B32: Equations for complex functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 12
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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