Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations
By D. C. Barnett, R. Halburd, W. Morgan, R. Korhonen. Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 2007
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It is shown that, if $f$ is a meromorphic function of order zero and $q\in\mathbb{C}$, then \begin{equation} \label{abstid} m\bigg(r,\frac{f(qz)}{f(z)}\bigg)=o(T(r,f)) \tag{\ddag} \end{equation} for all $r$ on a set of logarithmic density $1$. The remainder of the paper consists of applications of identity \eqref{abstid} to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions of $q$-difference equations. The results obtained include $q$-shift analogues of the second main theorem of Nevanlinna theory, Picard's theorem, and Clunie and Mohon'ko lemmas.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/6835115475c9928b6036333f318017795893a71a
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Holomorphic curves with shift-invariant hyperplane preimages (2009, methodology): “Theorem 6.1 can be proved by finding q -analogues of Theorems 2.1 and 3.1 and adapting the proof of Theorem 1.1 suitably, where the q -difference analogue of the lemma on the logarithmic derivatives from [3] is used in place of Theorem 5.1.”
- Clunie theorems for difference and q‐difference polynomials (2007, background): “Similarly, as the finite order solutions play a key role in complex difference equations, solutions of order zero are in focus for complex q-difference equations; see the introduction in [3].”