Finite-order meromorphic solutions and the discrete Painlevé equations

Abstract

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.

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

“…In fact, (1.9) fails to hold even for the simple entire function f (z) = e e z and η = 2πik, (k = 1, 2, 3, · · · ). After this paper is completed we learnt 3 that R. G. Halburd and R. J. Korhonen [18] have also obtained an essentially same estimate (1.10), and its interesting applications in [19] and [20]. Although our problem regarding (1.9) is somewhat weaker than the (1.8), we show that it is already sufficient for our applications and, more importantly, we shall show by examples that both the upper bounds and the finite order assumption are the best possible.…”

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

“…For a proof of the relations (46)- (48) see any standard reference in the Nevanlinna theory, for instance [54], for (49) see [111], and for (50) see [50]. In addition to (46)- (50) the identity…”

“…Recently, there has been increasing interest in applying Nevanlinna theory to study meromorphic solutions of complex difference equations [20,21,48,58,67,80], and in particular, to detect integrability in discrete equations [1,49,50,111].…”

“…Instead of just repeating the proof of theorem 5.4 here, we take the opportunity to apply the method used to prove theorem 5.4 in [50] to single out the difference Painlevé III equation out of a natural class of difference equations. To this end, assume that w is an admissible meromorphic solution of (58).…”

Entire functions sharing sets of small functions with their difference operators or shifts

“…Recently, a number of papers (including [1,3,4,[7][8][9]11,13,15,17]) have focused on value distribution in difference analogues of meromorphic functions. In a recent paper [15], considering Theorems A and B, Liu investigated the cases when f (z) shares sets with its shift f (z+c) or difference operator ∆ c f := f (z+c)−f (z), where c is a non-zero constant, and proved the following Theorems C-E. …”