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arXiv · 2108.06085

A Malmquist--Steinmetz theorem for difference equations

Yueyang Zhang·Risto Korhonen
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Abstract

It is shown that if the equation \begin{equation*} f(z+1)^n=R(z,f), \end{equation*} where $R(z,f)$ is rational in both arguments and $\deg_f(R(z,f))\not=n$, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term $R(z,f)$ takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case $\deg_f(R(z,f))=n$ of the equation above and thus provide a complete difference analogue of Steinmetz' generalization of Malmquist's theorem.

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Yueyang Zhang, Risto Korhonen. 2021-08-13. A Malmquist--Steinmetz theorem for difference equations. https://arxiv.org/abs/2108.06085

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