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

Abstract

The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlevé property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z + 1)w(z − 1) = R(z, w), where R is rational in w with meromorphic coefficients.

On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions

“…We adopt a more flexible interpretation on the Bachmann-Landau"big-O" notation [38, p. 11] so that for a complex function f (z), f (z) = O(ψ(r)) is interpreted throughout this paper to mean that there is an r 0 > 0 such that |f (z)/ψ(r)| < K holds for some K > 0 and for all r = |z| > r 0 . Recently, there has been a renewed interest in difference and q-difference equations in the complex plane C ( [2]- [6], [8]- [10], [14]- [18], [21], [23]- [24], [26], [35], [36]), and in particular, Ablowitz, Halburd and Herbst [2] proposed to use the Nevanlinna order [19] as a detector of integrability (i.e., solvability) of non-linear second order difference equations in C (see [2], [15], [17], [9]; see also [33]- [34] and [11, pp. 261-266]).…”

On the value distribution of some difference polynomials

“…Recently, a number of papers (including [1][2][3][4][5][6][7][8][9][10][11]14,15,17,18,21,22]) have focused on complex difference equations and difference analogues of Nevanlinna theory. Bergweiler-Langley [2] first investigated the existence of zeros of △f (z) and △f (z) f (z) and obtained many profound and significant results.…”

On Properties of Finite-Order Meromorphic Solutions of a Certain Difference Painlevé I Equation

“…Halburd and Korhonen [11][12][13] used value distribution theory and reasoning related to singularity confinement to single out difference Painlevé I equations from the difference equation (1.1). They obtained that if (1.1) has an admissible meromorphic solution of finite order, then either w satisfies a difference Riccati equation, or (1.1) can be transformed by a linear change in w to some classical difference equations, which [3] A certain difference Painlevé I equation 465 include the difference Painlevé I equations 4) where a, b and c are constants.…”