The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equations. These results include a difference analogue of the Clunie lemma, as well as other results on the value distribution of solutions.  2005 Elsevier Inc. All rights reserved.

If f : C → P n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation τ (z) = z + c, then f is periodic with period c ∈ C. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartan's second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both of which are proved here. Finally, an application to the uniqueness theory of meromorphic functions is given, and the sharpness of the obtained results is demonstrated by examples.2000 Mathematics Subject Classification. Primary 32H30, Secondary 30D35.

Let w(z) be a finite-order meromorphic solution of the second-order difference equationwhere R(z, w(z)) is rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else equation ( †) can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painleve equation of the form ( †), together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.

It is shown that, if f is a meromorphic function of order zero andfor all r on a set of logarithmic density 1. The remainder of the paper consist of applications of identity ( ‡) 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.

This research is a continuation of a recent paper due to the first four authors. Shared value problems related to a meromorphic function f (z) and its shift f (z + c), where c ∈ C, are studied. It is shown, for instance, that if f (z) is of finite order and shares two values CM and one value IM with its shift f (z + c), then f is a periodic function with period c. The assumption on the order of f can be dropped if f shares two shifts in different directions, leading to a new way of characterizing elliptic functions. The research findings also include an analogue for shifts of a well-known conjecture by Brück concerning the value sharing of an entire function f with its derivative f .

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For n ∈ N, the n-order of an analytic function f in the unit disc D is defined bywhere log + x = max{log x, 0}, log + 1 x = log + x, log + n+1 x = log + log + n x, and M (r, f ) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equationwhere the coefficients are analytic in D, satisfy σM,n+1(f ) ≤ α if and only if σM,n(aj) ≤ α for all j = 0, . . . , k − 1. Moreover, if q ∈ {0, . . . , k − 1} is the largest index for which σM,n(aq) = max 0≤j≤k−1 σM,n(aj) , then there are at least k − q linearly independent solutions f of ( †) such that σM,n+1(f ) = σM,n(aq).Some refinements of these results in terms of the n-type of an analytic function in D are also given. Mathematics Subject Classification (2000). Primary 34M10; Secondary 30D35.

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