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On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

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Abstract

We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People’s Republic of China

    Yik-Man Chiang

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

    Shao-Ji Feng

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  1. Yik-Man Chiang
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  2. Shao-Ji Feng
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Correspondence to Yik-Man Chiang.

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Dedicated to the eightieth birthday of Walter K. Hayman.

This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (HKUST6135/01P).

The second author was also partially supported by the National Natural Science Foundation of China (Grant No. 10501044) and the HKUST PDF Matching Fund.

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Chiang, YM., Feng, SJ. On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane. Ramanujan J 16, 105–129 (2008). https://doi.org/10.1007/s11139-007-9101-1

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