Ilpo Laine

Complex Difference Equations of Malmquist Type

In a recent paper [1], Ablowitz, Halburd and Herbst applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. A typical example of their results tells us that if a complex difference equation y(z + 1) + y(z − 1) = R(z, y) with R(z, y) rational in both arguments admits a transcendental meromorphic solution of finite order, then degyR(z, y) ≤ 2. Improvements and extensions of such results are presented in this paper. In addition to order considerations, a result (see Theorem 13) is proved to indicate that solutions having Borel exceptional zeros and poles seem to appear in special situations only.

Uniqueness of meromorphic functions sharing values with their shifts

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 at least three values counting multiplicities 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. More precisely, it is proved that if c 1 and c 2 are complex constants which are linearly independent over the real numbers, and f (z) is any non-constant meromorphic function sharing three values counting multiplicities with the shifted functions f (z + c 1) and f (z + c 2), then f is an elliptic function with periods c 1 and c 2. This can be seen as a new way of characterizing elliptic functions.

Differential polynomials generated by linear differential equations

This article is devoted to considering value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. In particular, we consider normalized second-order differential equations f″+A(z)f=0, where A(z) is entire. Most of our results are treating the growth of such differential polynomials and the frequency of their fixed points, in the sense of iterated order.

Remarks on Complex Difference Equations

Halburd and Korhonen have shown that the existence of sufficiently many meromorphic solutions of finite order is enough to single out a discrete form of the second Painlevé equation from a more general class f (z + 1) + f (z − 1) = R(z, f) of complex difference equations. A key lemma in their reasoning is to show that f (z) has to be of infinite order, provided that degfR(z, f) ≤ 2 and that a certain growth condition for the counting function of distinct poles of f (z) holds. In this paper, we prove a generalization of this lemma to higher order difference equations of more general type. We also consider related complex functional equations.

Tropical Nevanlinna theory and second main theorem

We present a version of the tropical Nevanlinna theory for real-valued, continuous, piecewise linear functions on the real line. In particular, a tropical version of the second main theorem is proved. Applications to some ultra-discrete equations are given.

Uniqueness of Entire Functions and Their Derivatives

This paper is devoted to studying the uniqueness problem of entire functions sharing a small function with their linear differential polynomials.

An oscillation result of a third order linear differential equation with entire periodic coefficients

We prove that the periodic equation admits a solution with finite exponent of convergence if and only if where n is a non-negative integer satisfying a certain (n + 1) × (n + 1)-determinant condition. Moreover, we obtain explicit representations for such solutions. Our result is somewhat similar to a result due to Bank, Laine and Langley [5] for a second order equation.

Locations of values of meromorphic solutions of algebraic differential equations

Meromorphic solutions of algebraic differential equations has been intensively investigated during the last three decades. These studies were mainly concerned with the growth of solutions and numbers of their a-points. In the present article we transfer to study location of a-points of meromorphic solutions for some important classes of algebraic differential equations. To this end, we apply the properties of proximitly and comparability of a-points. These properties offer additional information to the classical value distribution theory by describing locations of a-points of arbitrary meromorphic functions. In particular, we show for first-order algebraic differential equations that mutual locations of different a-points of the solutions are completely determined in terms of the equation.

Tropical versions of Clunie and Mohon'ko lemmas

This article presents versions of the Clunie and Mohon’ko lemmas in the max-plus semi-ring setting (also called tropical setting) of meromorphic functions. The main tool to prove these results is the tropical version of Nevanlinna theory.

Factorization of some entire functions

In this paper we prove, under certain conditions, that the entire functions of the form are prime, except perhaps for an exceptional set of a-points of logarithmic capacity zero. Here a, is complex constanth is a transcendental entire function and P is a non-constant polynomial.