Kazuya Tohge

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.

Holomorphic curves with shift-invariant hyperplane preimages

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 �(z) = z +c, then f is periodic with period c 2 C. This result, which can be described as a difference analogue of M. Greens Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartans 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.

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.

Riccati Differential Equations with Elliptic Coefficients

In this paper, we consider the Riccati differential equation w′ + w2 + a℘(z) = 0, where ℘(z) is the Weierstrass ℘-function satisfying (℘′)2 = 4℘3 − b, b ≠ 0. Assuming a = (1 − m2)/4, m ≥ 2, m ≠ 6n, we first show that all solutions are meromorphic, investigating their periodicity and doubly periodicity.

On Shared-Value Properties of Painlevé Transcendents

In this paper, we study some shared-value properties of the first Painlevé transcendents by applying its distinctive value distribution. One of the results is verified also for the second and the fourth Painlevé transcendents.

Meromorphic functions which share the value zero with their first two derivatives

G. Jank, E. Mues and L. Volkmann proved that if a nonconstant meromorphic function f shares a nonzero finite value a CM (counting multiplicities) with its first two derivatives f′ and , then f≡f′. It is also noted there that this is not the case for a=0. In this note we consider the case a=0 and IM (ignoring multiplicities) under certain andiiional conditions, one of which requires that the third order derivative should also share 0 IM with and the other is on the number of the zeros and multiple poles of f. We prove that each of the conditions can reduce f/f′ to possibly a linear polynomial

On Meromorphic Solutions of Linear Differential Equations with at Least One Transcendental Coefficients

Applying a result obtained in [TK], we investigate the zero-distribution of fundamental solutions of some linear differential equations with at least one transcendental coefficient, and compare the results with some of the well-known results on equations with the polynomial coefficients. Results corresponding to Frank’s theorem [F] on exceptional fundamental sets and a theorem due to Frank and Hellerstein [F-H] on non-homogeneous equations are obtained.