Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

By R. Halburd, R. Korhonen. 2005

609 citations recorded by the source corpus.

The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary dierential equa- tions. In this paper, a dierence analogue of the Logarithmic Derivative Lemma is presented, and then applied to prove a number of results on meromorphic solutions of complex dierence equations. These results include a dierence analogue of the Clunie Lemma, as well as other results on the value distribution of solutions.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c97b9570cb449a9253556fa9508b3a4a1572b5d5

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In Applied Mathematics

1. R. Korhonen
2. Zong-Xuan Chen
3. R. Halburd
4. Jilong Zhang
5. X. Zheng
6. I. Laine
7. Y. Chiang
8. Shaoji Feng

In Geometry and Topology

1. Zong-Xuan Chen
2. R. Korhonen
3. R. Halburd
4. I. Laine
5. X. Zheng
6. Y. Chiang
7. Shaoji Feng
8. Ajweezi. aljali

Citations in the corpus

• On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions (2006, background): “Halburd and Korhonen established a version of Nevanlinna theory based on difference operators [16], Bergweiler and Langley [6] considered zeros of difference operators, and Ishizaki and Yanagihara [26] developed a difference version of Wiman-Valiron theory for entire functions of small growth.”
• Clunie theorems for difference and q‐difference polynomials (2007, methodology): “In [8], Halburd and Korhonen proved difference counterparts (see [8, Theorem 3.”
• Some properties of meromorphic solutions of q-difference equations☆ (2010, methodology): “[27] J. L. Zhang, R. Korhonen; On the Nevanlinna characteristic of f(qz) and its applications, J. Math.”
• On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane (2006, result): “Korhonen [18] have also obtained an essentially same estimate (1.”
• A Difference Picard Theorem for Meromorphic Functions of Several Variables (2009, methodology): “Halburd and the author used one-dimensional difference analogues [15, 16] of some of the main results of Nevanlinna theory to prove that the existence of at least one finite-order meromorphic solution, which is not simultaneously a solution of a first-order difference Riccati equation, is enough red”
• Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations (2007, methodology): “Recently, similar tools involving shifts have been developed to study ordinary difference equations [6, 13, 14].”
• Finite‐order meromorphic solutions and the discrete Painlevé equations (2005, methodology): “In the remainder of this subsection we state a number of recent results on difference equations and Nevanlinna theory [18].”
• Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity (2009, background): “Chiang and Feng have obtained similar estimates for the logarithmic differences in [3], and this work is independent from [9,10].”
• Nevanlinna theory for the difference operator (2005, background): “1 in [4] the error term on the right side of (2.”