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Malmquist-type Results for Difference Equations with Periodic Coefficients

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Abstract

An earlier result by N. Yanagihara leads us to consider the nature of a meromorphic solution \(f\) of a difference equation

$$\begin{aligned} f(z+c)=\sum _{j=0}^n p_j(z)(f(z))^j, \end{aligned}$$

where \(p_j\) are periodic entire functions of the period \(c\in \mathbb {C}\), \(p_n \not \equiv 0\) and \(n >1\). We shall show that if \(f\) is non-periodic entire and of finite order of growth, it must be algebraic over any field that contains the coefficients of the difference equation. We also consider the special case \(n=2\) more carefully and obtain specific information on the solution \(f\). Our methods are based on Nevanlinna theory and algebraic field theory.

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Acknowledgments

I thank Professor Gary Gundersen for the discussion and the motivation for the topic. I also would like to thank the referee for the idea of simplifying the Proof of Theorem 4 and for other improvements.

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Authors and Affiliations

  1. Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101, Joensuu, Finland

    Jarkko Rieppo

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  1. Jarkko Rieppo
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Correspondence to Jarkko Rieppo.

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Communicated by James K. Langley.

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Rieppo, J. Malmquist-type Results for Difference Equations with Periodic Coefficients. Comput. Methods Funct. Theory 15, 449–457 (2015). https://doi.org/10.1007/s40315-015-0109-z

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  • DOI: https://doi.org/10.1007/s40315-015-0109-z

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